I'd rather combine game-balancing aspect of missions not lasting too long with fluff explanation of nations being forced to deal with the threat on their own. I assume, as much as questionable from the tactical point of view, aliens having near limitless resource sin comparison with Earth, expendable troops of generally stable morale (because of their role as a part of a hivemind) - they can afford loss of a single ship here and there, unless it's a target of great strategic importance.
At the same time nation getting squad of them on their backyard would be probably reasonably disturbed, with military on ready and going in as quickly as it's confirmed Phalanx isn't going in anytime soon. They'd probably want to act quickly - risk of infiltrators, horrible PR if aliens will be allowed to start a carnage in some major population center and access to the technology for themselves would be quite an incentive.
It would need to be balanced, though as it's nearly surely would lead to situations where having several downed aien ships, unless going back and forth without rest, player has to ignore some of them. Getting noticeable happiness hit for each such vessel ignored could be a problem, but also a good reason to at last add not only damage to crashed aliens but a chance to take some of them outright (granted, I hope for such mostly because of fluff - I always wondered how come, if the ship itself got broken into pieces, there are absolutely no alien casualties, no corpses near the wreck etc).