For what it's worth, my therapist felt pretty genuine. One thing that helped is to actually ask about his day, and how he's been. I think he was surprised I cared.
Another thing that helps is that BetterHelp makes it effortless to try out someone new. Maybe we've been lucky, but me and two other people haven't needed this feature; we all stuck with the first one we matched with. But it's there if you need it.
I am a member of a Christian Health Sharing ministry which is not insurance, so they work with my therapists to share the costs. My mental and medical health costs are shared by other Christians in a community of mutual support, rather than being bankrolled by insurance company premiums. And all I need to do is arrange "self-pay" terms with the provider and have the bills sent to the ministry!
Even though it is not insurance, health sharing ministry members are exempted from the tax on Americans who do not carry health insurance. It's a pretty sweet deal; my monthly shared amount is about half of the premiums for a "catastrophic" Obamacare plan found on the market.
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.