Stripe's problem is that they make it look like anyone can accept payments with just an API integration, but the financial system does not work like that for many complicated reasons. As long as you call them up, get an account manager, and build that trust like any business relationship, you'll be fine. The issues come if you sign up in the same way that a bad actor would, in the lowest-touch way.

I'm not sure Square or Braintree or anyone else is much different in this regard

I have no involvement with stripe, or OP, but I will strongly advise you: DO IT! Im sure many people who complain about stripe are probably slightly unsavory (not saying OP is), but seriously, the day shit hits the fan, you are simply out of luck unless you somehow have a gazillion twitter followers, or enough luck/clout to get the attention of some strip guys here.

Consider your options carefully!