Sophia here from Shopify.
The third-party payment gateway you choose does need to be one of the supported gateways listed.
What country is your store located in? If there's a gateway you wanted to use that isn't on the list for your region, I'd be happy to submit feedback about that to the appropriate teams here. I can't guarantee when and if that gateway would be made available, but it's important that our teams know which gateways you'd like to use so that they can consider potential integration going forward.
Are you selling exclusively online, or do you sell in person as well? Outside of the credit card gateways, you may also be able to set up alternative payment methods like cryptocurrency, or manual payment methods like bank deposits or cash on delivery.
Please let me know if you have any questions.