... and they give the licenses out to almost everyone...
Speaking specifically for Texas eligibility, that not quite correct. For starters, any conviction or deferred adjudication in the previous five years of any crime more serious than a traffic ticket, any felony ever in your lifetime, or behind on child support, is disqualifying, among other things. Truthfully, LE agencies have looser restrictions on who they can arm than the state takes with LTC applicants and license holders. Think about that for a second. There are LEOs who are disqualified from being issued a LTC.
Once issued, there are recurrent state and federal background checks that continue to verify eligibility. In the in-between time, the state revokes at the time of conviction. Lautenberg agreement also adds a revocation layer.
The Texas legislative history from the inception of LTC has is filled with subject applicants and license holders being held to a heightened level of scrutiny, continuously monitor and enforce, and in return a few additional privileges are provided. That’s bookended with enhanced penalties and disqualification when a license holder violates the terms and conditions (laws) for the privileges granted.
As shown elsewhere, the state tracks this data very closely, aggregates it at the county and state level, and makes it publicly available. Like any process, it’s not foolproof, but it works. A lot better than most people expect.