Hero likes his hand and makes a bit bet/raise on flop.
Villain has some kind of draw and does not like to fold. 'Etiquette' in this game is to either RIO or RIT but to always stick to it, so villain knows what you'll say if the money goes in.

If you're RIO, villain will peel.
If you're RIT, villain will re-pot and ask to stick it all-in and run it.

This happens absolutely all the time. RIO guys get peeled one more street way more, and (sometimes) get villains to fold on bricks. So even if your equity when/if the money goes in doesn't change, the fact that people will play so massively different against you has to alter your overall equity a lot.

I could even see a case for RIO in deepstacked games to be less variance. (less huge pots at 60/40, more turns and rivers with playable SPRs, where you can exercise your edge and your hand-reading)

This is true. I win many pots on the turn without showdown bc v knows he doesn't get 4 cards to beat me by shoving a draw on the flop.

I don't always get to stack him, but he doesn't get to realize his equity. Not willing to take the time to do the math if any exists, but that very well might equate to lower variance.

Sir thats delusional thinking

Dont worry math doesn't exist in poker

Not sure why you would say that. Another poster was more eloquent in his explanation above. Players might peel a card on the flop if they know you run it once but fold on a brick turn and not realize their equity on the river.

I'm only saying that, based on experience, when I changed my policy a while ago to only running it once in my regular game (3-5 sessions per week), I am noticing that I am getting more fold turns and less flop shoves from random draws. While that's not great when I have dominating draws, I am more than happy to play medium strength hands and bluffs in position armed with that knowledge.

As the other poster articulated, that might actually cause less variance in certain situations.

If am am being delusional, please expound as to why you think so. I hate to immediately dismiss the comment, but without any explanation, I don't know how else to take it.

Thank goodness, I thought I might have been missing something.

Always run it once.

This is a rare time where your willingness to gamble gives you an edge.
There is a huge psychological factor of losing your stack when going all in. (This is a large part of why shortstackers exist.)

This fear from my deepstack opponents gives me fold equity on my bluffs. This fear dissuade my ball-less opponents from shoving with draws. This fear limits my villains.

I have a wider range of moves to apply to more situations than my fearful opponents, which ultimately leads me to dominate them.