I have not seen the movie but what I have read about it indicates there are some illogical aspects to it.
As I recall, the mission is to escort a dying Cheyenne & family from the southern plains, probably in The Indian Territory (modern Oklahoma) to his former homeland in Wyoming or Montana. That should be a distance of at least 500 miles as the crow flies.
The logical route would be to travel to the nearest train station and take a train hundreds of miles east, transfer to a train going north for hundreds of miles, and then take a train going west for hundreds of miles to a station close to the destination. That way, the characters might have to ride only two hundred miles instead of at least 500 miles and they would get there faster and more comfortably.
I note that the fictional date is said to be 1892, and that was a very late date to encounter hostile Comanche warriors on the warpath. The hostile Comanches surrendered and went to the reservation at the end of the Red River War in 1875. About 170 Comanche warriors and their familes left the Reservation in The Buffalo Hunters War of 1876-1877. And I think that after 1877 it was almost impossible to be attacked by hostile Comanche warriors.
So setting the movie in 1872 instead of 1892 would make it much more plausible to encounter hostile Comanche warriors. And there would be fewer railroads in the west in 1872, so the nearest stations to their fort and to their destination might be much farther away, thus reducing the advantages of using the trains and making riding all the way less silly.
I also noted that in the clip the flagpole at the fort has two flags flying. I didn't get a good look, but assume the top one was the US flag, and doubt whether any other flag would be on the same flagpole, even under the US flag.
I note that you seem to be assuming that it was normal for US soldiers who enlisted as privates to be promoted to higher ranks. That was not the case in those days. The normal term of service was 3 years, and only a minority of soldiers would reenlist for a second term. The majority of enlisted men were privates in their first and only term.
The organization of a cavalry company varied over time, but usually consisted of a captain, first lieutenant, second lieutenant, first sergeant, six sergeants, four corporals, a few specialists like smiths, two muscians (trumpeters) and a varrying number of privates, usually less than 100 - and the ranks were usually far from full, so that a company on campaign usually had 40 to 60 men available for duty.
So if a five man escort consisted of an officer, a non commisisoned officer, and three privates, nobody would think that there were too many privates. I noticed that when they left one of the soldiers had a first sergeant's insignia, so his company would be without its first sergeant until he got back.
If the average private served for 3 years, about half the privates would have over 1.5 years experience and about half the privates would have less than 1.5 years experience. And it is quite possible that any soldier with more than 0.5 years experience would not be considered a raw recruit. I think that statistically an average private might have one combat with Indians during their term of service, so an average private wouldn't get very experienced in Indian combat.
And my guess is that a greenhorn like Dejardin was chosen to be the natural choice for the noncom to assign the most menial chores to. Remember that in real life in 1892 the odds against meeting any hostile Indians and having to fight wwould be almost 100 percent, so Dejardin's lack of fighting experience almost certainly would not be expected to matter.
If, on the other hand, the colonel was expecting major trouble in the immediate future (highly unlikely in the real 1892), he might have decided to send Private Dejardin away so he wouldn't be a weak link and so one more experienced solider could be kept at the fort.
In the movie Only the Valiant (1951) Captain Richard Lance is assigned a suicide mission to defend his fort by regarrisoning the sadly misnamed Fort Invincible wiht a small group and holding off a horde of Apaches as long as possible. Lance selects a group of soldiers he considers to be drunken, cowardly, insane, or otherwise bad soldiers, explaining to each of them why he think that the main fort will be better without them to weaken its defense.
And it is possible that if the colonel was expecting a major uprising (which would be highly improbable in the real 1892) he might think that the fort would be better off if Dejardin left the fort on the escort mission instead of a more experienced soldier.