I made a video tutorial for a simultaneous-four-player version of Backgammon called Quadragammon, invented in the 1970's. It's loads of fun, and also works for three players. Give it a try if you have friends (By the way, I've successfully taught Quadragammon to loads of people who've never played backgammon and they loved it regardless!)
Compared to a chouette, how would you rank this variant in terms of fun/engagement/skill/strategy etc.?
Unfortunately I have very little experience with chouettes (been only playing backgammon for a year and mostly against XG), but compared to the 3-4 chouettes I played in a tournament, I find quadragammon more fun because everyone is directly involved and playing in the game, rather than only making cube decisions and being prohibited from giving their opinion on the current teammate's checker plays (as was the case for the chouettes I played).
It also introduces new dimensions of thinking, such as how a move might be beneficial for you but leave your teammate hanging (un-makes a point you had with your teammate, leaving his checker as a blot), so you have to analyze with him carefully whether doing this makes sense or not. And the bear-off process is considerably more dangerous. Firstly, in quadragammon, if you're bearing in on the bottom right for example, the opponent right in front of you (the player who bears off on the top right) comes back from the bar on your bear-off quadrant, just like in regular backgammon. But then aditionally, your other opponent (the one on the bottom left)'s back checkers must cross your bear-off region in order to get to their home board (your bear-off quadrant is his quadrant with points 7-13). This creates more bear-off excitement compared to backgammon.
One disavantage of quadragammon is that by only having 8 checkers rather than 15, situations where you only have one legal move arise more often, due to the higher probability of other moves being blocked. However, when there are multiple viable decisions, they tend to be very interesting, which offsets this disadvantage.
Quadgammon looks like a really fun variant. For me it seems like a game I'd play with people who don't normally play backgammon. I feel like its a variant that I can get family members to finally play with me. To me though, it seems like it'd be a lot of fun! I know I'm gonna try and get people to try it with me.
I second what GGG said. I want to try this! Good on ya for attempting a revival!