You would typically integrate with a gateway who integrates with one or more banks who integrate with the schemes to get global coverage.

There are generally a number of domestic gateways in each region (I’ve worked with gateways in Australia, Malaysia and a couple of other places) but there are also larger providers.

There are name brands like Braintree and lesser known but still quite large, like Adyen. The API quality is extremely variable.

Checking out multiple gateways is a good idea. Stripe is supposed to be great but IIRC is relatively expensive.

We used CyberCash (I think) and Authorize.net, which were to Stripe as Altavista was to Google search.

That is to say that even 20 years ago there were middlemen in the payments space, they just were pretty horrendous to work with.

There were, and still are, many gateway providers with APIs for payments. Authorize.net, NMI, USAePay, BrainTree, Payeezy, even Paypal was common enough once upon a time. And those are just a handful that I can list off the top of my head. (I know that some of those I listed have acquired each other recently as well.)

I guess it depends on what counts as handling payments yourself. Stripe can handle just about everything to managing subscriptions and emailing invoices and more. At least nowadays. It’s a lot more advanced than systems that roughly just provide an API to manage payment methods and charge them, like Authorize.net for example.

Also, handling the abstraction between multiple kinds of payment methods (cryptos, different card networks, PayPal, etc.) may also be a considerable amount of complexity that a payment vendor can help with.

I have never directly interacted with the financial system at its lowest levels. I’ve heard it is quite a trip.