Authorize.net was (and still is) a "gateway" - you pick the merchant and use Authorize.net for the actual technical layer of processing a transaction - which was kind of a confusing concept, but there were plenty of merchants on authorize.net's marketplace that had much better rates than Stripe. I don't recall there being an onerous application process or the restrictions you mentioned, but it could be that I was shielded from that side of things.
Both Authorize.net and PayPal had pretty gross API's (and documentation) but they were pretty solid once you got them working. Obviously it depends on the nature of your business, but saving even 10 cents per transactions goes a long ways for many and quickly justifies a day or even a week of additional coder time. It wasn't that onerous of a task to implement one of the non-Stripe options.
Paypal of course was it's own beast with it's insanely draconian policies towards account holders, but I never had to deal with that (luckily).