https://moodle.org/plugins/availability_paypal
https://moodle.org/plugins/enrol_authorizedotnet
https://moodle.org/plugins/enrol_bitcoin
https://moodle.org/plugins/enrol_classicpay
Well having been 'looking for days' perhaps you've already considered and rejected these that I found in a single search that took about 15seconds. But if so, that would be nice to know so we don't waste such huge amounts of time doing such searches for you!
There are many places using Moodle with a paid gateway model, but you are quite right it is not its default model and relies on plugins and adaptations to do that. There are several around, along with the ability to adapt, tweak or even write from scratch, your own authentication/enrolment plugins.
You could even use a separate pay-to-play kind of database and then use the built in automatic external database enrolments feature to enrol/unenrol users as required (which would be my preferred way to achieve what you seem to want).
From my knowledge there are also integrations that work with Wordpress and other systems to allow a 'course catalogue' to be advertised and course enrolments to be bought.
Moodle tries hard to provide the options it needs within core for its CORE areas. Other requirements are then handled by plugins, developed by users with needs outside the core and often shared with the community, and by the nature of Moodle as Open Source software, allowing you to develop it to match your specific requirements.
You may also want to check out the whole area dedicated to business uses of Moodle https://moodle.org/course/view.php?id=32 to see if anyone there shares your experiences or can offer advice for you.