If you recall, before X-COM came along noone had anything but the standard rifles, which are horrifically ineffective, and instead of using the best and brightest exclusively like X-COM does, they have a regular military, or in some cases, just a police force.
The best and brightest... I like to imagine the soldiers X-COM recruits to be in the vein of the recruits for forum LPs, both because it's funny, but also makes a perverse sort of sense. X-COM looks for two qualities in their recruits.
One is expendability. Especially when starting out, X-COM squad casualty rates are atrocious, as they get thrown into battle against forces outside their ability and comprehension.
The other is a lack of respect for human life. Yes, X-COM is dedicated to defending the Earth against all alien threats. But that's the high level. When boots hit the ground, if a civilian gets caught in the crossfire, then a civilian will have to die in the crossfire. Because to fight the alien menace with the force and ruthlessness required, X-COM soldiers can't afford to avoid collateral damage or non-soldier injuries.
There's also the third quality, born out of the first two, that X-COM recruits tend to be unhinged pyromaniacs with foolhardy bravery. Since survivability and public relations are out of consideration anyway, this is as useful a quality as the others.
In light of all that, combined with way you have to support yourself selling alien artifacts and manufacturing weaponry, it's easy to see how X-COM evolved into the competitive private service company it appears as in Apocalypse. And why by then, nobody except the government likes them either.