Am I rational to be hopeless to have an academic career after my mental recovery?
(this answer is based not on my personal experience, but I know a number of people struggling with depression, some of them also in academia, and some quite close to me)
I don't think it's rational to decide that your academic career is hopeless. You are on a PhD track after all. You are still in a position where you can do good work, make people notice you, and ultimately get a good career going. Especially in Germany my feeling is that where you do your PhD is not very indicative of how well your career will be going afterwards - I know professors graduating from the smallest universities that I have never heard about in other contexts, and I know plenty of graduates from TU Munich that went nowhere after graduation.
What you need to do - and I am aware that this every difficult - is to let go of what happened in the past and focus on the future. How can you bring yourself into the best position? If your advisor does not have too many contacts you will need to work on these contacts yourself (honestly, this is nothing particularly unusual - this is the situation that at least 2/3 of PhD candidates find themselves in, not every advisor is an international superstar). If you can't get a postdoc via reference, keep your eyes open for posted positions. Go to conferences, meet people, make people be aware that you are or will be on the job market. And even if your academic career plans indeed do not work out, there will still be plenty of other opportunities for a PhD holder to pursue. Don't get married to one specific career path, and don't tie your self-worth to whether you make one specific thing work or not.

That said, the elephant in the room is that you also need to take care of yourself. Self-treating a depression with &quot;sports&quot; is dangerous - maybe what you are actually doing is distracting yourself without actually working on the underlying problems, and this has a tendency to only work for so long. I strongly recommend you to give therapy another shot. You don't necessarily have to agree to take medication (although, as I said in a comment above, not all anti-depressants make you tired, but it is true that pretty much all of them potentially come with some problematic side-effects), but some therapy or treatment is long-term really the only way forward. That said, you are perfectly in the right to be picky when choosing what psychologist / psychiatrist(s) to work with - many are frankly very bad at their job, and even the better ones may be a bad fit for you personally.
Again, your focus needs to be looking forward how to improve your life, not looking backwards at what went wrong before. This isn't easy for anybody, and I know that depression only makes it harder to focus on a positive outlook rather than things that went badly in the past. A lot of this has to do with acceptance - accepting that you have in fact a medical condition, and that you will need to do and deal with some things that people without that condition don't have to bother with (and some of them may indeed be impacting your ability to perform in a workplace). We accept this to be the case with non-mental medical conditions, and it's also true for depression.
Ultimately, you need to remember that long-term your health is more important than your career. You can certainly still have both, but if you decide to ignore your health because of your career there is a good chance that 10 years from now you have neither.