Hello,
itch.io admin here. It's not as simple as providing a button for a developer to add there. itch.io provides entire end-to-end integration as a store with purchase verification, file access levels, reward tiers, refunds, support, fraud, etc. For us to add a new payment provider we need to provide this level of integration into our system. We don't think it's a good idea to let a game developer to a button to something we aren't integrated with because consumers may not be aware that they aren't getting the same level of support when they select that payment method.
If you want to integrate with a third-party payment system or store then I recommend you add a link to that in the description of the page. That way it isn't part of itch.io's UI for checkout and there is no confusion about where the purchase is taking place.
I hope that explains things.
(Regarding other payment providers, we do want to add more, but it can be a large overhead for us regarding payouts, taxes, fees, etc., so it takes time, thanks)