The therapists I know do certainly work with every day people struggling with (perhaps existential) anxiety/fear/depression. Where those concerns are negatively impacting someone's life, they are real and pressing concerns - if common.
These therapists also regularly work with clients that are dealing with the effects of being victims (or perpetrators) of various forms of physical, sexual, and physiological violence. Even in wealthy areas, people do terrible things - particularly to the weak and the young.
That job can be very intense.
I think the unvarnished therapist response to existential questions is something like: that’s fine, those are valid, but they do in fact have no answer; the only way out is to turn our attention to the day to day problems of living.
If you keep harping on unsolvable problems, I think they’ll mostly feel annoyed.
That said, if your therapist was "destroyed" by your unsolvable but grave questions - you should just move onto someone else. That isn't likely to be a common occurrence.
Most therapists I know agree that there are many grave questions which are not "solvable" in therapy.