It can be a bit annoying to face and then move to Right/Left Shoulder but there is good reason for it.
What the TC 3-21.5 Says
Chapter 5, paragraph 5-2, bullets 5 and 6. (Emphasis mine.) This is for the Army, Air Force, and Space Force. It also encompasses AROTC, AFROTC, AJROTC, AFJROTC, SFJROTC, CAP, and related programs.
Facings, alignments, and short-distance marching movements are executed from Order Arms or Sling Arms. When the movement has been completed, automatically return the carbine to Order Arms.
Facing movements are executed from Order Arms or Sling Arms. When a Facing is necessary to establish the direction of march, the Facing movement is executed before the command for the manual of arms. After a marching movement has been completed, Order, ARMS or Sling, ARMS is commanded before the command for the facing movement.
What MCO 5060.20 Says
Here is a quote from Chapter 3, Section 1, paragraph 1. While the bold emphasis is mine, I included the rest of the text for a wider explanation. This is for the Marine Corps, Navy, and Coast Guard. It also encompasses NROTC, MCJROTC, NJROTC, CGJROTC, and related programs.
c. On the command “FALL IN,” if armed with a rifle it will be positioned at order arms.
d. Facings, alignments, and short distance marching movements are executed from order arms. Side step, back step, open and close ranks, and close and extend are short distance movements. Forward march may be given from order arms to march units forward for a short distance. When these movements are commanded while at order arms, it is necessary to come automatically to trail arms and step off simultaneously on the command of execution for the movement. The rifle is returned to order arms upon halting.
e. Before a command for any marching movement (other than the executions stated in paragraph 1.c. of this section) is given to armed troops, they are faced in the direction of march and their weapons brought to right (left) shoulder, port, or sling arms by the appropriate command. After a marching movement has been completed and it is desired to execute a facing movement, the command to order or unsling arms is given, followed by the command for the facing movement.
f. When at a position other than sling arms, the troops must come to port arms for double time. When the troops are in formation, the commander gives the appropriate commands.
But Why?
No explanation is given as to why facing movements cannot be performed at Right/Left Shoulder, but we can use some reasoning and most likely discover why.
The inability to perform facing movements at Right/Left Shoulder can be attributed to safety and formation integrity. When individuals fall into a formation, they should maintain their dress, cover, interval, and a distance of 40 inches all around. However, this is not always the case due to the varying heights of individuals in the ranks. When a formation is properly sized, safety is less of an issue, but it is still important to play it safe by facing the direction of march, hoisting the rifle to a shoulder, and carrying on. Obtaining Right or Left Shoulder beforehand would not be wise because it could compromise the safety and integrity of the formation.
When marching in formation at either shoulder, executing flanks and “To the Rear” movements is different because a formation often enlarges when marching. This enlargement provides more space and flexibility for such movements, reducing the risk of collisions or accidents.
In Summary
Facing movements cannot be performed at Right/Left Shoulder due to safety and formation integrity and not “because we’ve always done it that way.”