You know, I think the stakeout thing could work like this:
You have X hours of no frills preparations, getting the team briefed and ready - it's just padding, to let most of those interested to participate but favoring those with a position nearby with the extra time.
Then you have some more hours of covert insertion preparation - if you launch before this, you start with your cover blown since you just tossed your guys out of a black helicopter a'la EXALT or whatever in an urban centre rather than being sneaky. This allows a whole stealth system thing for groups which choose to specialize in it along with an option for Michael Bay Ops not to bother.
Then you could have the final period where you can wait to boost your odds... somehow. My first idea was everyone who joined starts at the same time, but the heavy preppers get some free moves for every Y hours, but I feel like that might cause issues on second thought.
Or maybe fluff it as recon, and give them things like getting a general shape of the area, number of enemies, position on detectors, etc., with increasing amount of that stuff revealed every so much time spent. So the 'locals' get a home turf advantage by knowing the lay of the land.
Speaking of that last thing, it could be expanded; you could have an overmap structure/deployable of, say, controlling the local SIGINT agency that give you sight range for battlescape camera objects, and a Hacker class that can do that temporarily for single structures in battlescape.