I for one don't particually like the cold War aesthetics.
As for why the aliens aren't using the heaviest units from the get-go? Could be different reasons. Perhaps they are a scouting force and have limited resources themselves? Perhaps more advanced units need time to bring in from wherever they are coming from? Perhaps those advanced units ARE there from the get go, but deployed in more pressing theaters. Xenonauts certainly wouldn't be a big enough threat early on to warrant such resource expenditure, especially since the alien wouldn't really know who the Xenonauts are or the extent of their abilities.
If this is the case, the game should comminicate it trough messages, briefings, news reports, etc.. Make sure to not just mention, but also show in some way the world armies ARE fighting.
Something like:
"Commander, I'm sure you already read the reports about the crushing defeat of the 5th army corps? The alien invaders are deploying weapons and troops significantly more advanced than what we have encountered so far. Fortunately, they appear to have a limited number of them, and we can consider ourselves lucky our men haven't run into them yet. However, sooner or later it is bound to happen. I urge you to divert funding into weapon development to combat these threats. Trying to capture samples would be of great benefit, but bear in mind the risks involved."