What's a bit newer is that it's a solved problem for people in other countries too; 10 years ago or so I really wanted to process payments, but none of the easy payment processors supported my country (mostly Braintree and Stripe back then), so I had to go with eWay (awful) and a merchant account from my bank (double awful). The company I'm with now (same country) uses Stripe, and it's a massively better experience. (Not being a Stripe fanboy; they have some very strong competitors, I just happen to be familiar with Stripe.)
Keep in mind: It matters where you (the merchant) are. Accepting payments from someone in Africa when you're in the US is trivial; accepting payment from someone in the US when you're in Africa is much harder. You can see where Stripe is supported here: https://stripe.com/global (and most of their competitors are in the same or fewer markets).
Also, there's still a lot of pretty bad legacy payment processors around, and they're often a fair bit cheaper. So if your company has been accepting payments online for twenty years, it may still be a pain for you, I wouldn't know. :)
But yes, the mom-and-pop website in Topeka should be using Shopify or have paid a web shop to throw together something equivalent using off-the-shelf packages, and it should be using Stripe (or Braintree, Square, Adyen, whatever), and then yes, they can accept payments globally, and the experience will be really nice. It's not even any extra work; it just comes out of the box for "free".
If people in those places don't have credit cards, I'd wait for Stripe to add whatever payment method they do use. (They've already added systems like Alipay and Klarna that are mainly used outside the US.) In the meantime, they can email me and we'll work out if they can mail me a check or use Western Union, or maybe I'll just give them an extended free trial or something. There are probably not so many people in the world who (a) don't have credit cards and (b) are interested in my software-as-a-service that I wouldn't be able to manually help them on a case-by-case basis.
(Disclaimer: I have worked on a mom-and-pop e-commerce website, but it shipped physical products and didn't operate internationally.)
This is an honest question, because I'm starting to seriously question my beliefs (I honestly though it was difficult.)