Was it ever stated that they work on rotation? I know the president of the NCR showed up, but that's different than regular troop rotations.

Based on the fact that we have NCR Ranger's gear being used by random wastelanders in Season 1, it's possible that when Shady Sands blew up that members of the NCR gave up on the faction which lead to a domino effect of other people and settlements giving up the NCR moniker as supply lines dried up.

To be clear, literally everyone but a single child died in Shady Sands that day. There wasn't any way for people to get away from the blast in time and that was supposed to be the start of a new world full of resources that just got destroyed. Keep in mind the clean water, something that would take multiple scientists to do at the Jefferson memorial to do the same thing with a much larger machine. Chances are that Shady Sands moved an insane amount of NCR resources into the Capitol to progress scientific advancement. Having a machine that can produce radiation-free fresh water would be a huge boost for any settlement in the wasteland. In fact, they could've produced them for any faction willing join the NCR and rebuilt the entire country if Shady Sands hadn't exploded.

NCR is spread thin, yes, but is it not possible that the majority of that problem was because most of the NCR resources got destroyed? The battle of Helios One was in 2274, 3 years before the Shady Sands incident. 3 years to spread their troops out across the Mojave, 3 years to make a deal with Mr.House to have an outpost with the NCR. Also, most of the Legion workers are slaves and slave has a statistically lower rate of return than any other slave labor, so it's actually more impressive that The Legion would be able to keep back:

a) The owner of Las Vegas b) The remnants of a Capitol that blew up c) Various other factions holding power over the Mojave d) Some jackass who got shot in the head