Not to toot my own horn (I mean the app is pretty simple) but it's a splinter that's been bothering me about macOS for years and years.
For three finger swipes I'm using a trick to synthesize them without private API, but for other swipe types I'd need to synthesize them using the private API... However there haven't been too many requests for other event types thus it's not very high up on my TODO list.
(I'm the author of BTT)
Also BTT allows you to bind them to many many predefined actions in addition to keyboard shortcuts, e.g. "Trigger Menubar Menu Item" which can be very powerful.
I'm basically using CGEventCreateData to save an original system three finger swipe, then I'm just generating an event based on that saved data and refresh the timestamp and mouse location before sending it.
Wish developers were legitimately able to generate those three-finger swipe events, because they are ubiquitously supported and really useful. (Just the other day, I discovered that Preview allows you to switch pages by swiping up and down! Weird.)