You should remember that, most nations have the military academy that give the commissions for the graduate. And while they lacks the experience, they have enough knowledge to run the armed force, that is not required for the enlisted(that is expected to take care of themselves to smaller unit).
Actually the way how make an officer in Battletech universe is already suit for your taste; even if they graduate the academy, they starts with sergeant, NOT O-1 grade officer as our worlds does, and they have to prove themselves on field in order to actually get the commission and begin the life of the officer.
Since officers in our worlds often requires bachelor's degree(other than field promotion of dire needs), and some nations at least grants specialist for the person with bachelor's degree when they join the armed force, so it is nothing wrong to give them the sergeant by default. A full four years is enough to earn the rank of sergeant when they choose to join the armed forces starting to private. Not all will does, sure, but someone with enough talent will does, and it is no wonder that such valuable individuals are enter the arcademy and not getting dropped. And it is a waste of time to put such person(that surely complete the advanced course for the soldier) bottom of the line; they should take the position much better than the privates. Also, despite of their lack of experience, they must able to deal with their 'subordinals' or they don't have enough right to be called for a good officer either.
The roman empire was the good example of this. Their elected commander that commands the unit that is similar to our battalion level, was usually the startline of the career of the young noblebirth without real experience for the battle. But, despite of their lack of experience, since no soldiers are want to die in vain under the incompetent commander they only vote for the most able candidates as their own commander. So those young nobles are have to study very hard for the art of war, and distinguish themselves to be actually elected. And, despite of their no real experience, centurions are able to make the gap.
Also the important factor is armor branch. A tank commander is a sergeant. A battlemech is a biped tank. And... is it possible to rise from the rank of private, while keep up the title or battlemech pilot(and individual battlemech commander)? Imperial Japanese in WWII already proves that using private or specialist pilot results catastrophic failure. Even for the tanks, if all the tank commander in a platoon is at least sergeant but one have the rank of private, it is something gone horribly wrong.