Very much agree in terms of the best strategy being simply to start a plain-jane Kapitalgesellschaft, probably either UG or GmbH, and go from there.
It is indeed possible to have a so-called "kein-Mann GmbH" where the company has bought back all of its shares, though my understanding is that it's a bit of a legal grey area, and certainly not settled law.
I agree that the legal infrastructure is almost certainly there, at least in the sense that there are absolutely plenty of lawyers and lawmakers that specialize in this area. My point is simply that, because it's so much less common, basically every situation ends up being unique, which means that the work that the lawyers are doing is almost always a one-off, which makes it really expensive. And so it's just not worth the effort.
This stuff actually gets a lot of attention from lawmakers: For example, in the U.K. you have the “CIC” (community interest company). Some 13 years ago, David Cameron tried pretty hard to motivate enthusiasm for the idea of a “third sector”, something that's not government and not for-profit. In the U.S. you also have the “L3C” (low-profit limited liability company). In Germany, you have the idea of “Verantwortungseigentum” which was on the agenda for the previous government, though they then didn't get around to it, and you had Sahra Wagenknecht making it into a big talking point for her campaign.
But I don't think a lack of legal infrastructure is really the limiting factor here: As you noticed, we do have foundations (Stiftung) of various types, as well as coops (Genossenschaft). In addition, regular partnerships (like KG, OG) have recently been opened up so that their bylaws can now prescribe a purpose that isn't for-profit. A club (Verein) which in and of itself isn't for-profit, can have a sort of dual identity because it can become the proprietor of a sole proprietorship with a for-profit purpose (at least I seem to recall reading that such a thing is possible). Oh, and, of course, a corporation can, in theory, own 100% of its own shares. I recall reading about that, just please don't ask me where. You can basically wipe out ownership that way, without being subject to the stringent rules around foundations.
So, legal structures are as powerful and flexible as they are underutilised: I think it's the psychological side that explains why.
Usually, even if you have very good intentions, your best bet initially is to start your entity as a for-profit. Being able to operate cheaply and without cumbersome decision-making structures beats lofty aspirations for any business that just gets started. Not turning a profit in a given year (and not paying taxes because you don't turn a profit), is an option you always have. (There's no special paperwork needed for that). In fact: Not having any profit to worry about when it comes to your structure is the likely outcome. Having a profit and trying to decide how to make it so that your profits won't corrupt you in your idealism is a problem you would quite like to have! (Again: From the perspective of a founder who is just getting started).
Then, the day comes where you turn a profit quite regularly. And, at that point, once the flywheel has got going, truly giving up control will be psychologically difficult.