In our various MechWarrior campaigns since I was a JROTC and later ROTC cadet, we came up with our own kinda explanation and work-around to try to explain the strangeness that is MechWarrior privates, etc.
So, we took the idea of Law Enforcement Rank, where in some jurisdictions promotion from the patrolman ranks to Sergeant is one track, but one can also promote to Detective. Detective Sergeant, then, is a higher rank than a beat cop Sergeant.
So, MechWarrior Private is a Private in the sense of "Private Soldier," but much more "Gentleman Ranker" in the Napoleonic sense. A member of the societal upper crust waiting for an opportunity to slide into an officer billet. So while an infantry Corporal might seem to outrank a BattleMech Private, that is not functionally the case as MechWarriors are considered to be a senior service. This might be like the discussions my retired USAF friend and I had over the push to re-instate the Warrant Officer ranks into the USAF. There are pilot officers who are pushing to bring back USAF Warrants because they want to fly, not command, and as was noted upthread expressing a desire for such is a career-limiting move. They argue that pilots who wish to concentrate on flying would be better off as Warrant Officers so as to avoid all the mandatory bonus fun that is leadership training and other professional military education requirements.
So, we have in our Regiment in the fiction an enlisted track parallel to, but explicitly above, the other arms of the unit. The insignia is similar, but with a 'Mech silhouette identifier inside the chevrons. This is Master MechWarrior, E-7. Sort of like the old Spec-5 through Spec-7 (-9?) ranks.
This seems to iron out the wierdness of Academies producing Sergeants, if that Sergeant is explicitly senior to those in the more mundane combat arms by virtue of being a Mechwarrior Sergeant. It also doesn't overload the TOE with too many by-God officers, or a crapton of Warrants.
I mean, if your formation is mostly warrants, you don't have a formation. You have dozens of pilots who disappeared to get coffee and were never seen again until Drill Weekend is over.
But seriously, given the idea that each 'Mech is like a Knight, and has a retinue of techs, astechs, etc. to support it, the idea that a MechWarrior is basically a Warrant Officer with an enlisted entourage fits both the social cachet of the setting and the officer/enlisted balance of a modern military better than having to jump through the mental hoops of MechWarrior Privates. The system we use in our games is the same thing, just under a different name, and minus the salutes and the "sirs/ma'ams" and "misters/misses" for every MechWarrior.