We also built our Stripe integration using only the testing tools they provide. The first real payment in production was completed by our product guy, but only as a sanity check. Maybe we got lucky, or maybe Stripe is just that easy to integrate with, but we haven't been surprised by any production behavior.
A "sanity check" is a test. Alway always always test that real prod payments end up in the bank account they're supposed to before whoever can rollback goes home for the day. _Always._ And document the existence/process/details/outcome of that "sanity check" test.
Not documenting the existence/process/details/outcome is exactly the distinction that I intended to make when I called it a "sanity check". I was really just offering a data point, not a comprehensive guide to best practice.