You generally need to be a bank or regulated financial institution.
SWIFT is just a messaging layer for correspondent banking. If you're not a bank, SWIFT access is about as useful as suddenly learning your country's president's cell phone number: Definitely cool, but the two of you would probably not have much to talk about.
When something really screwed up you resorted to sending MT599 messages around, free format text messages.
I believe it's all SWIFT over IP this century so probably not dissimilar to any other API you might use.
I.e. if you go and create a neobank through a baas provider you will inadvertently get access to SWIFT, it won’t be direct but you have access.
To get true access you have to go through and insane process and have a ton of controls and policies and create trust with the governing body of SWIFT.