All residents in Germany need to have a registered address. The reasons for that are numerous:
- no need for a separate voting registration,
- it gives a place to deliver legal notices,
- it helps enforcing tax regulation (though tax law has another definition of "residence"),
- certain taxes are assigned to the municipality where you are registered,
- no need to do a utility bill or credit report dance when trying to prove your address to third-parties,
- it was always done like that.
The last is reason enough as far as bureaucracy is concerned.
As far as your question goes, there are a few wrong assumptions.
First, nobody is trying to limit or deny registrations. Municipalities actually get money for every resident they have registered through tax allocation.
Second, if you (semi-)permanently live at some place you are not considered "officially homeless" but as someone has didn't properly register themselves (which could result in a fine).
Third, multiple people can be registered at the same address. How else could people register themselves in huge apartment buildings? Or even a family living together? If there is a suspicious amount of people registered they might want to check whether other regulation (such as minimum space per person) is adhered to.
Fourth, to register somewhere you need proof that you actually live there. This is where you probably ran into problems. This proof is usually a letter given by the person allowing you live at a place. Note that this is not necessarily the owner of the place but literally the person allowing you to stay (the authorities can check with the owner though). Often subletters won't sign that piece of paper, either because they think they are not allowed to or they don't have permission by the owner to sublet the place to begin with. If that happens you are supposed to tell that to the authorities (§ 19 Abs. 2 BMG) who then can choose to fine the one providing the place for refusing to sign the paper. (The need for proof was recently reintroduced after it was noticed that there was a huge amount of people who registered at an address where they didn't actually live.)
Fifth, if you never have been registered before you can't actually properly pay your taxes as you need a tax ID for that. This is automatically assigned when you – surprise – register for the first time. If you are not an EU/EEA/CH citizen you also need to register yourself before they can change anything about the residence permit within Germany.
To end with an answer to your question if taken literally: German citizens can register themselves as homeless (ohne festen Wohnsitz) if they really are. For foreigners this is a bit more complicated. EU citizens usually don't have freedom of movement rights if they can't properly support themselves, a living space is supposed to be part of that. Other foreigners likely violate their residence permit.