After all this time, I still haven't finished my game.
Once I got Laser Weapons, things went a bit more smoothly. It also helped that after some more Mutons and Gazers I finally hit a period of mostly Floaters and Waspoids, which are a pushover compared to the first two.
By the time the Mutons returned, I was much better prepared.
For a while, there was relatively smooth sailing, and I was continuously making progress. Quite a few nations retreated from Xcom, but I think my actual funding never dropped.
I exterminated all the cults, captured some juicy UFOs and even a few bases/colonies, finally made it to Turbolaser and then to Plasma Weapons. I researched (I think) all vehicles, tried to deal with all the frankly a bit overdone side-theaters (underwater, zombies, UAC, cyberdudes) ... and then everything just grind to a halt.
In the last 6 months or so, probably more, I only had, besides UFO grind, a vampire queen mission (following research suggested I need to retrieve some mcguffin, but no mission spawned yet), a moon landing (following research didn't really amount to anything significant) and the capture of some sort of extradimensional battleship.
For about three months, nothing at all out of the ordinary happened. I have 250 researchers twiddling their thumbs and mechanics working hard on my fourth Avenger as well as some more flying armors ... and no idea what to do next.
I'd expect some serious moon mission to further the story now, but as I wrote, nothing seems to materialize.
What could be the problem?