Technically you should never wear a US flag on your left (neque ad sinistram). Throughout history your arms (weapons) were deployed on your right side and that is the side you "presented" to your opponent and having your colors on the wrong side might be confusing in battle for those fighting with you. This is where we get the term "right hand man" (the person who defends you in a fight) as does (traditionally) a wing man fly off to the right.
Yeah, I wouldn't be so sure about that. Especially since the wearing of flags on shoulders is a pretty recent thing versus a cockade on a hat, giant banners, painted shields, facings or just different colored uniforms being among the many distinctive markers of a soldier on a battlefield before the 20th century. Plus, if is was such a big tradition the majority of countries don't follow it to include our very tradition-minded Commonwealth colleagues who wear the flag on their left shoulder along with most other countries that wear flags on their uniforms, with a few having them on both shoulders and but even fewer doing the right sleeve thing like the US Army.
Doesn't the army wear the flag on the right shoulder because they have a specific use for the left shoulder patch?
Their unit patches (usually division or regimental) go on the left. They've been very consistent about that for a long time- and the unit patch predates shoulder flags. I gotta say, I respect that consistency and tradition.
The 'shoulder sleeve insignia' is official term for the distinctive unit patches in the Army, all soldiers they wear their current unit/formation on the left sleeve but those who were in 'combat' with a unit can also wear their SSI on the right sleeve. Several Army types I've worked with use the term 'slick sleeve' when talking about those without a 'combat patch' on their right sleeve, usually derisively if they are mid-to-senior ranked folks. 'Combat patch' can be a bit of a loose term, I've known folks who have deployed to Kuwait and Qatar and have gotten 'combat patches'.
Since a lot fewer folks prior to 2001 had 'combat patches' it made some sense to put the flag on the opposite sleeve from your SSI to give the uniform the appearance of balance, particularly greens, but it is a lot less so now with so many soldiers with 'combat patches'.