.22LR trainers have been in military use for about as long as there have been .22LRs.
Francis Gary Powers (and I would assume the other U2 pilots) had a High Standard .22lr as part of their survival pack.
The biggest problem with combat use of the .22LR, besides it LOW power, is that the rounds themselves are relatively fragile. You can get a full auto .22LR to run, fairly well (with good ammo) BUT other than as a pampered specialist weapon its simply neither very useful, nor good at surviving the rigors of infantry combat.
If an FA 10/22 or one of the cool belt fed "Browning" MGs jams at the range, its not life threatening. Pilots survival pistol, suppressed? good deal. Size & weight and bulk of ammo very good for the role.
For a ground combat role, including clandestine sentry removal, for the same amount of "effort" in weapon tech a much more powerful round is possible.
For example, a 9mm will do a better (more reliable) job taking out a sentry than a .22LR, both suppressed so noise is not a factor.
If you're thinking of something like a .22LR mini gun, the problem is twofold, one is the "return" on the investment, meaning how much downrange damage is possible for the size and bulk of the weapon, and the other is the technical issues of the heel type lead bullet of the .22LR being easily bent and not being well suited to high speed feeding, compared to rounds like the 5.56mm.