I really enjoyed the hell out of this game but agree that you (Goldhawk) made zero use of the 1979 timeframe, other than a few scattered lines in the doctor's reports (Internetwork, lol). Nothing would be significantly different if somebody went into the game files and changed 1979 to 2009.
Just look back at how many close-calls we had involving nuclear weapons, both real and fictional (Able Archer 83, Fail-Safe, etc.)...feels like you need to capitalize on that tension a lot more. In the game everybody just goes, "oh, aliens? Isn't that what Xenonauts was founded for? Cool, thanks guys, let us know when you win." Hawkish military brass on both sides probably would have read the initial unidentified air contacts and attacks as the handiwork of the other superpower and resulted in at least a brief shooting war. Nothing cataclysmic, but at least in the fiction I've read, authors like to ramp up the tension with a progression of initial provocation -> scattered air and naval aggression without actual shots fired -> isolated skirmishes -> one deadly battle -> nukes fly. Dunno how, if at all, that turns into an objective for the player to somehow prove the aliens are real or otherwise get the superpowers to look up rather than at each other.
Then, to upvote the idea above, both the USA and USSR should basically be vying for your complete attention. Each should be able to offer you something the other guy can't: usually the difference has been represented as the USA having a tech advantage while the USSR gets a sheer numbers advantage (which isn't strictly true but hey, that's the popular conception of the era). You've already had the Foxtrot being an adaptation of the high-altitude, high-speed MiGs which were designed to catch and kill U-2s and SR-71s, but if you take things in the direction of exclusivity, maybe if you piss the Soviets off, they won't give you any more MiGs? If you ignore the USA, you lose out on advanced radar and electronics, or stealth tech? I dunno, just throwing stuff out there. How you conduct your operations should also have some bearing on it; the Soviets don't care if you cause collateral damage and killed a bunch of civvies as long as you stopped that terror site, but the American media will crucify you for blowing up half of Anytown USA even if you did get rid of the aliens. And those opinions could obviously influence how much support you get, how many troops they'll make available to you, etc.
If you really wanted to take it to an extreme, maybe if you cuddle up to the Soviets, the Americans get so suspicious of what you're up to that they become outright hostile, and convince some of their NATO allies that you shouldn't be trusted, etc.
But, you interject, wouldn't an alien invasion make everybody be friends and work together? For my money, I doubt it. Even if you don't want to implement any of the USA versus USSR stuff, you've got to think that there would be opportunists elsewhere in the world who saw their big chance to take over that disputed border territory that's rightfully theirs, or those oil fields that their infidel neighbors really shouldn't be allowed to keep, etc. China thinks: you know what? I bet we could expend a hundred fighter planes to shoot down one of those UFOs, take their tech, find a way to beat them, and become the saviors of the world. That would show those Yankees and Rooskies who's really in charge around here.
TL;DR version: more Cold War politics/vibe, please.