How does applying for asylum grant the ability to "stay forever regardless of the outcome"?

Because once you're inside the country you will never get kicked out, unless you actively cooperate with the authorities.

To those interested in the actual data: On June 2023 roughly 280k failed asylum seekers were met with a a yearly deportation volume of around 16k or 6%.

Note that "failed asylum seekers" does, for the most part, not mean people actually resisting deportation in any illegal/violent way, but mostly consists of people who are being granted a sort of amnesty ("Duldung") for various reasons in line with german law.

This only leaves 54k people that would actually be candidate for deportation in June 2023, which increases the deportation rate to roughly 30%. At this pace deportation for the average failed asylum seeker that is actually traget of deportation shrinks to 3 years.

* https://www.dw.com/en/germany-migrant-deportations-rise-shar....

depends, at first the right to stay is time limited

I imagine that that's a relatively small percent of immigration, and should follow its own separate rules based more on humanitarian needs.