I agree in large part with what you've said here. Definitely feel like Operation End Game arrives far too soon.
I also liked the early scripted story missions, but feel like atm the Cleaners are more of an afterthought than a credible threat to your existence. Sure, the Cleaner Base has been beefed up to the point of likely taking some of your early soldiers out with it before you close them down for good, but other than whittling away at your funding governments' nerves with the increased monthly panic threats, they don't really do much other than sit there. Feel like they really need to be fleshed out in greater detail ... right now it's far too easy to track down the main base and put an end to them. I'd suggest actually having the Cleaners conduct some terror missions of their own in different locations across the globe, to give the monthly panic threats some actual merit. I'd also suggest adding some false trails/missteps along the way, that could end up in ambush situations, but would then provide clues to help triangulate the actual base if you survived those trials. A potential cleaner assault on your initial base when you are on the verge to discovering their location could also be .... fun?
I also agree on the objective/research/interrogations/UFO wreckage exploration bit. I actually love the tactical combat missions, it's my favorite part of the whole game, and I'd love to see some more game mystery unveiled by doing more of those ops.
Agree on the long stretches, at least on Veteran and higher difficulty, anyways. Maybe after the first few months, the aliens get smarter about direct contact with the countries you've got bases in and instead vigorously go after countries you have little or no presence in, while keeping you occupied in your base areas ... for instance, barraging you with light alien fighters and interceptors in your base countries, blowing up airliners, strafing airfields and oil rigs, etc and also threatening your troop transports when you send them out to tactical missions (something that rarely happens now) thus tying up the few fighters you've likely managed to afford to make defending home turf/protecting your troops while sending out more deadly, demoralizing bombing runs on countries further away from your bases, giving you a real chance to lose some of them over time.
Alternatively or in addition, regarding the long stretches of nothing, this could be broken up by local forces requesting Xenonaut assistance in places that you do occupy, putting out less disastrous alien incidents to garner some small bit of panic reduction or perhaps something more useful (like minor monetary rewards). These could either be frequent enough to not all be doable, or sporadic enough to just add a little hint of extra danger/flavor to the game.
I still haven't gotten to the endgame w/o a new experimental build coming out to distract me, so I can't comment on that bit, lol ...