This is actually a pretty tricky question. The standard black hole solutions are highly idealized in order to make them tractable. In particular, they have a lot more symmetry in time than you would expect a black hole to have. Actually forming a black hole from realistically distributed matter undergoing gravitational collapse is a more complicated problem to talk about in GR. So you pose a genuinely interesting question.
The basic idea is that singularities are a very general feature of general relativity, not confined to specific solutions to the field equations but sort of inevitable for a large number of initial conditions. So while the standard black hole solutions are highly idealized, we have good theoretical reason to believe that singularities exist in non-trivial solutions to the Einstein Field Equations.
You're confused by mathematical terminology. Since mathematics is ignorant about time, "inevitable" is a synonym to "exists" there, because mathematics can easily assume infinite time passed by usual induction, but in physics time isn't negligible. Even in terms of mathematics you need to believe in actual infinity to believe that formation of a black hole completes.
I think the singularity theorems address it however: https://en.wikipedia.org/wiki/Penrose%E2%80%93Hawking_singul...
The basic idea is that singularities are a very general feature of general relativity, not confined to specific solutions to the field equations but sort of inevitable for a large number of initial conditions. So while the standard black hole solutions are highly idealized, we have good theoretical reason to believe that singularities exist in non-trivial solutions to the Einstein Field Equations.