Probably because I'm a hardcore Legion fan.
If a few lads and lasses can save the day then the professional police, military, and rescue forces of that era, which have access to the exact same powers in larger quantities, should be able to do the same.
I have to admit that I've lost track of what is and isn't Legion Canon through thirty plus years of reboots and retcons but wasn't the original conceit of the group that they were a government-sanctioned peace keeping team?
The Legion was originally bankrolled by R.J. Brande, then eventually the United Planets took over the majority of the funding.
As for why the various worlds that sent Legionaires didn't just form a giant police force made up of those worlds, one can easily assume that there are all sorts of diplomatic and procedural issues in the 30th Century under which the UP might not have been able to even if they wanted that.
Plus Legionnaires aren't assigned to the LSH. They're individuals who chose to try-out (there was a draft early in the post-Zero Hour reboot version, but it was abandoned). You may have a planet full of people with "super powers," but who's to say they want to venture out and police the galaxy? And even amongst their own people, the Legionnaires are extraordinary. Titan might have been populated by telepaths, but few if any have Imra Ardeen's skill or power.