I think you have in part answered your own question, but let me enhance that a bit.

Democracy is, if not the least efficient form of government, certainly amongst them. It is impossible to operate swiftly if everyone is casting votes all the time, and everyone needs to be polled before taking an action.

More importantly, no military including the U.S. undertakes desirable things, and there will always be more GIs than officers. So every engagement would not start (at least on the part of the democratic army), because the more populace group of enlisted personnel would always outvote the officers, and no one with any degree of self interest would ever vote to put themselves in harm's way. I am not trying to impugn the honor or courage of any servicemen and women ever, I am simply saying that their desire to go home whole to their families, and to see their comrades in arms do exactly the same, will naturally outweigh their desire to take risks.

That is, unless they are ordered to do so. A military chain of command places increasing responsibility and knowledge at each level up, and it is (hopefully) this increased knowledge that makes the decisions more prudent as one goes higher along the chain.

A chain of command is highly efficient at reacting to information in a timely manner, even when that information is critical in nature (this may have been less true in eras before electronic communication, but even then the speed of everything was lower). In this it is the opposite of a democracy. Informed, well-educated democracies are ponderous, slow things that are heavy with inertia, but the hope is that the decisions they make are the best ones overall and in the long term (I am talking about ideal systems). Military command is rapid, quick to react, but at least at some level inappropriate for the long-term goals of society building, perhaps because of their very rapidity of action.

Democratic decision making is slow and inept. This is why the Military is not a democracy. We don't have a democracy, either. We have a democratic republic. It is faster than pure democracy and a little less influenced by the herd, but it is a far cry from Military efficiency.

There hasn't been true democracy since ancient Greece.

I think you have asked an interesting question on whether or not an effective military organization can be a democracy. And, I think the only possible answer is that it has never been tried with an organization as large as the present day US military.

Representative democracy does not mean that soldiers would take a vote before every task, such as before doing "push-ups". It would mean that they would elect their officers and they would need to obey their officers for the duration of their term.

There have been many cases of effective military units where the officers were elected and sometimes where the officers could be removed based upon a vote. One of the famous examples being the Caribbean pirates which operated as a democracies. American Indian military units were very effective and acted as democracies. Some left wing revolutionary military units have operated at the unit level as democracies. In many early United States militia and regular army units the officers were elected.

Perhaps, one particular problem with a democratic military would be that in some of the past wars the soldiers would have voted to end the war early... but, perhaps that would have been a good thing.

So, I do not think it is impossible, but rather it has never been tried.

The United States military's purpose is to defend the United States it has nothing to do with defending the existing political structure.

Straight from G.K. Chesterton (What's Wrong With the World)

If a camp is surprised by night
somebody must give the order to fire; there is no time to vote it.
It is solely a question of the physical limitations of time and space;
not at all of any mental limitations in the mass of men commanded.
If all the people in the house were men of destiny it would
still be better that they should not all talk into the telephone
at once; nay, it would be better that the silliest man of all should
speak uninterrupted. If an army actually consisted of nothing
but Hanibals and Napoleons, it would still be better in the case
of a surprise that they should not all give orders together.
Nay, it would be better if the stupidest of them all gave the orders.
Thus, we see that merely military subordination, so far from resting
on the inequality of men, actually rests on the equality of men.
Discipline does not involve the Carlylean notion that somebody
is always right when everybody is wrong, and that we must discover
and crown that somebody. On the contrary, discipline means that
in certain frightfully rapid circumstances, one can trust anybody
so long as he is not everybody. The military spirit does not mean
(as Carlyle fancied) obeying the strongest and wisest man.
On the contrary, the military spirit means, if anything, obeying the
weakest and stupidest man, obeying him merely because he is a man,
and not a thousand men.

The two main values (IMO) in military culture are discipline and obedience. They help the military machine run smoothly. Often decisions need to be made quickly and will not please a lot of people. For the sake of expendiency and to preserve loyalty in a way the military needs to have a hierarchy with little room for dissent or opinions. A "democratic" army would be the most disorganized, ineffective army in the world.
And if anything, I think in the military you can't afford to have a marketplace of ideas the same day you can in a democracy because of how necessary it is to have unquestioned loyalty. It is actually counterproductive to have disagreement.
That said, it is fundamental to have civilian oversight for the military. That way the military's power does not go unchecked and you prevent situations like in several Latin American countries where the military has actually become an obstacle to consolidated states and democracies because it becomes an actor as powerful or more powerful than civilian government.

A democracy serves to protect the interests of all the parties within it, by giving them all a voice. Theoretically, group decisions respond to all these voices and come up with a best answer that all can live with. In practice, of course, the majority is frequently not concerned with protecting minority interests, so democracies normally have to be moderated by some other force. In our case, we have a constitution and a judiciary.

The military does not exist to best serve its members, however. The members of the military have sworn an oath to protect a society which is largely separate from the military. A democratic approach would not reflect this much larger stakeholder in the affairs of the military. In our society, the military only exists to protect and serve the civilian population, who wouldn't get a vote.