Hansi Flick's side beat Costa Rica 4-2 in their final Group E match at Al Bayt Stadium, but that was not enough to see them through to the round of 16.
Japan's surprise 2-1 win over Spain ensured the Samurai Blue finished top and Luis Enrique's side secured second place due to having a superior goal difference to Germany.
Having failed to advance from their group in Russia four years ago, Germany were left stunned after falling short once again in Qatar.
A shock 2-1 loss to Japan in their opening match cost Germany, who held Spain to a 1-1 draw before claiming a first victory of the tournament in what proved to be their final game.
Bierhoff, Germany's national team managing director, said: "We're extremely disappointed. We're angry too because we had things in our own hands in all three games.
"Ultimately, football is a simple game. You have to score lots of goals and concede as few goals as you can, which we didn't manage to do.
"We could have put more pressure on Spain tonight if we had taken our chances and scored more goals. A trend of the last few years has been us throwing away games we’ve controlled."
Japan will face Croatia for a place in the quarter-final next Monday, with Spain taking on Morocco 24 hours later.
The former Germany striker warned time is up for the era that has seen top-flight footballers in Europe's top leagues securing huge contracts as a matter of course.
Since the coronavirus pandemic forced football to be widely suspended in March, clubs and leagues have warned of the impact of inactivity.
And even though many leagues are slowly restarting, behind-closed-doors games mean matchday revenue has been lost.
Bierhoff spoke of a "mass of mediocrity" within the leagues, and said such average footballers would have to accept they are not entitled to spectacular salaries.
"It has come to the time when many are ready to at least discuss it – even those who were not before the crisis," he told t-online.de. "Especially from the biggest clubs in Spain, England and France."
Bierhoff, now a director of the German Football Federation (DFB) said he would support salary limits that were "legally feasible and if we think about it together across Europe".
Football authorities are battling to protect the vital revenue stream of broadcasting deals, which have been the lifeblood of the game, allowing teams to finance an abundance of seven-figure salaries.
Bierhoff said: "I don't think that the top earners for fans are the problem – as paradoxical as it sounds. [Lionel] Messi, Cristiano Ronaldo or Jo[shua] Kimmich aren't the problem. The problem is the mass of mediocrity that swims in their wake and and are, in quote marks, 'overpaid'."
He added: "An average Bundesliga professional will have to expect that he will soon be offered less by his club."
The World Cup winner has been under huge pressure following last month's 6-0 Nations League humbling at the hands of Spain.
It ended a 12-match unbeaten run for Germany and was their heaviest defeat since losing to Austria in a friendly in 1931 by the same scoreline.
The German Football Association (DFB) last week brought an end to the speculation over Low's future by throwing their support behind the 60-year-old.
But Bierhoff has eyed up alternative options should Low leave the position he has held since 2006.
"You have to be careful with words. It is an ongoing process," Bierhoff told Sky Sport. "Anyone who thinks I'll only think about alternatives after the Spain game is insane.
"Of course, you always have something in your head. Joachim is always informed about my thinking - that's my job."
Asked specifically about former RB Leipzig coach and managing director Rangnick, who has long been considered a natural successor to Low, Bierhoff said: "I also spoke with him."
Jurgen Klopp is another to have been touted as a candidate after winning the Champions League and Premier League with Liverpool.
But Bierhoff ruled out waiting for Klopp's Anfield contract to expire, saying: "There is no plan to hire Klopp in 2022."
Low guided Germany to World Cup success in 2014, four years after finishing third, and has reached a European Championship final and two semi-finals without winning the continental competition. His team also triumphed at the 2017 Confederations Cup.
Bayern Munich chief executive Karl-Heinz Rummenigge said he has not seen enough of Low lately, adding he must "deal more aggressively with the situation", but Bierhoff defended the coach's record in charge of Die Mannschaft.
"Low has had one bad year in 16 years," he said. "It is difficult to drive development under the current conditions. He has energy, conviction and a clear vision."
He added: "I don't see Rummenigge's statements as being that critical. Everyone has their own opinion. The important thing is that you are convinced of your path.
"The one who carries the ultimate consequences is the coach. Joachim does that. There are many examples in which the public opinion has also been different.
"Our team is inexperienced and needs time. Our greatest talent, Kai Havertz, has played only 10 international matches.
"We need to find some regularity but that has not been possible because of so many injuries.
"Other teams, like the Netherlands, have needed six years to redevelop their side - they missed two tournaments while doing it."
Another hot topic of debate surrounding the Germany national side regards the futures of Thomas Muller, Jerome Boateng and Mats Hummels.
All three players were dropped by Low last year and have not been recalled since, but their fine club form and the defeat to Spain have increased scrutiny.
"Jogi has never once said the door is completely closed to them," Bierhoff said. "Nothing personal happened. There is certainly no stubbornness.
"I'm happy that they are all doing so well. Joachim will ask himself, 'Who are the players I can work with best?' It is not a matter of principle."
The 2014 world champions are searching for a new coach to succeed Joachim Low, who will leave his role after the Euro 2020 finals later this year.
Flick, Low's former assistant, was predictably mooted as a potential replacement having enjoyed great success as Bayern boss.
The Bavarian giants won the treble last season and have already added the Club World Cup title in 2020-21, while defences of their Bundesliga and Champions League crowns remain on track.
Flick, who turned Bayern's fortunes around, is under contract until 2023, although club CEO Rummenigge will depart at the end of the year.
That will not spell the end of Flick's tenure, though, Rummenigge insisted last week, saying: "We are well advised to finish what we have agreed in the contract."
But this does not clash with Germany's plans either, as Bierhoff says the new man will not be poached from another job. Ralf Rangnick, who turned down a role at Schalke, has been linked.
"Nothing has changed for us," the team director said on Monday. "We always said we won't approach a coach who is currently under contract.
"I interpreted it in the way that [Rummenigge] wanted to back Hansi Flick and he did. That is the right thing to do as CEO of a club."
Bierhoff was facing the media at the start of the first international break of 2021, with the clock now ticking on Germany's recruitment process.
"We don't have limitless time, but we are not under pressure either," he said. "We have to appoint a new coach for the time after the Euros. We are going to take our time.
"The DFB delegation will be in the team hotel in the coming days. President Fritz Keller, Peter Peters, Rainer Koch and Friedrich Curtius will be there and we will discuss the situation.
"Obviously we will analyse, discuss and have long talks, and then we'll see."
Germany, who were expected to reach the latter stages of Qatar 2022, crashed out at the group stage, leading to the departure of team director Oliver Bierhoff after 18 years.
There was speculation boss Hansi Flick would follow him, but the former Bayern Munich coach confirmed he would be staying on board.
The new advisory group includes former Bayern CEO Karl-Heinz Rummenigge, Oliver Mintzlaff, and former Germany internationals Matthias Sammer, Rudi Voller and Oliver Kahn.
It will be led by Neuendorf and DFB vice-chairman Hans-Joachim Watzke.
"I think this is a group that knows a lot about men's football and the national team," Neuendorf said. "We are not an association in which one person decides everything.
"There will be a first discussion before Christmas. We will give ourselves a timetable there. We have to join forces towards the European Championship in 2024. It has to be a success."
Neuendorf also confirmed there will be a new group that will "take a close look at the entire business area for which Bierhoff was responsible".
This group will include DFB general secretary Heike Ullrich, former Germany international Philipp Lahm and European Championship ambassador Celia Sasic.
"We will see how we set up the area in the future in order to be successful," Neuendorf added. "And we will be self-critical."
Germany crashed out of the World Cup at the group stage, which led to team director Oliver Bierhoff leaving his role.
There was speculation boss Hansi Flick would follow him, but the former Bayern Munich coach has confirmed he will be staying on board.
Vogts, who led Germany between 1990 and 1998, believes the German Football Association (DFB) has paid the price for fostering a harmonious atmosphere and wants Matthias Sammer appointed to ruffle some feathers.
"I was surprised that Oliver Bierhoff went so quickly," he wrote in the Rheinische Post. "But it speaks for his character that he is retiring and thus taking responsibility after the second preliminary round exit at a World Cup in a row.
"But the problems of German football are not solved with the departure of Bierhoff alone. There are too many theorists at the DFB who are far removed from football. You have to be close to the game again; in player training, but also in coaching.
"I hope that Matthias Sammer will work in the background and support Hans-Joachim Watzke [vice-chairman].
"Matthias is a child of football, he knows and loves it, and he is a critical spirit who also addresses unpleasant things.
"That was missing in the DFB for too long. That's certainly something that Oliver Bierhoff has been accused of. I often pointed this out to him, but criticism was not welcome in his environment.
"That's why I left the DFB advisory board years ago. I hope Watzke will re-establish a healthy culture of debate in the DFB with Sammer in the background.
"There has been too much harmony in the DFB in recent years. This has led to a standstill in many places. Hansi Flick will also benefit from more friction in his work.
"It's right to leave him in office. It's not just with a view to the European Championship, but he has the full confidence of the DFB to put our football back on the road to success."
