Now that various versions of Commander are allowed at Regular, some of the players at my LGS believe they have found a loophole in the commander rules. I couldn't figure out where the error is, so I decided to post it here.

Basically, because the commander is considered to be part of your deck (CR 903.5a), and the sideboarding rules say that you can change the cards in your deck (CR 100.4), you should be able to change your commander between games in formats that allow sideboarding. (e.g. Tiny Leaders) In addition, you're supposed to play with the same deck in game 1 (MTR 3.15) but there's no rule that says that your commander has to be the same. So you could pull a different legendary creature from your deck in game 1, depending on who your opponent is. You could also change your commander in games 2 and 3 even if sideboarding isn't allowed, as long as that commander was part of your deck in game 1.

I don't think I'm missing anything, so I'm hoping that someone who knows the rules better than I do can find where I'm wrong.

There is currently no official support, either from Wizards or the Commander Rules Committee, for playing Commander (or variants thereof such as Tiny Leaders) in a multi-match tournament setting. As such, this is not something that the CR, MTR, or the CRC rules cover.

If your LGS would like to run such a tournament at Regular as a Casual Play event, they can do that, but they're going to have to answer these kinds of questions on their own.