I thought the logic in the OG X-COM was quite simple: The Aliens prefer stealth and subtlety, and they harvest Earth like it's a farm. They can't come and take Earth Independence Day-style because they don't have the forces for it. It's not a large fuck-off alien fleet, it's a small operation based in Cydonia, dependent on Elerium Supply from outside the Solar System and limited in numbers. They are not interested in engaging in total global genocide for the same reason a farmer doesn't go to his huge farm and slaughter every single animal at once. It's a harvest, not a culling.
Don't forget, they want to harvest DNA from every living thing on Earth, especially humans. And as we find out, humans are very much OP in the X-COM universe, the only race able to match Ethereals in psionic power, but without the whole "being husks that shouldn't even be alive" thing.
When X-COM becomes a thing, their go-to strategy is Terror Missions to intimidate human governments, attacks against X-COM's bases and infiltration/subversion/diplomacy to subvert human nations.
In the inferior Firaxis remake, the aliens seem to come Independence Day-style, but seem to operate otherwise much like their OG counterparts.
In the OG game, its very much a shadow war, and meant as such. Small teams of elite forces fighting each other in secret battles across the planet, daggers in the dark, Men in Black and G-Men, all very X(-COM)-Files.
Heck, in canon, the civvies didn't even know it was going on until after it was over.
The aliens only start killing people en-masse once X-COM loses and Earth's nations are compliant
(and then you get, well, the world of X-Piratez. Through in Piratez Earth seems less "Alien Colony" and more "Nature Preserve/Open Air Experimentation Camp". Like a zoo in which the animals are allowed to fight each other)