 It needs to be that if some random American decides "hey, I want to go to Cuba just for fun", they can do it.

If the US says you have to apply for a special license to travel for Cuba, but they give it to anyone who wants one, YES.

If licenses are restricted or hard to get, NO.

Does it need to be generally allowed, or is it enough if it's allowed with special permission or something? (Eg if 'tourism' or something similar gets added to the list 

At the moment US law prohibits US citizens from traveling to Cuba as tourists.

https://home.treasury.gov/policy-issues/financial-sanctions/faqs/topic/1541

(To resolve YES, tourism must be allowed by both US and Cuban law.)