First time I've EVER heard of it is Texas doing so, though you may be right Utah and Iowa have similar clauses.
Obviously being from Illinois, the biggest example I can think of at the state level was when Blago was impeached during his tenure as Governor (probably people from other states remember that one too, it was pretty shocking and unprecedented at the time).
There was ZERO talk of "temporarily suspending" Blago as Governor BEFORE the Senate has rendered a verdict, even though everyone KNEW he did what he was accused of, and he had been impeached overwhelmingly (117-1!) There wasn't even talk by the state legislature of finding a way to "suspend" him as Governor AFTER he did the unthinkable and actually DID appoint a U.S. Senator while under a "cloud of suspicion" for taking brides, and having been impeached for soliciting bribes for that very appointment. Rather, they tried to resolve the "problem" on the reverse end by simply refusing to seat Blago's 100% legally-appointed candidate, Roland Burris. The RATS didn't get very with that one (gotta hand it to Blago, it took some brass balls to do that!)
I agree that the federal government's system should have no affect on how state governments set up their own methods, but there's a reason why nobody has bothered to "temporarily suspend" a single federal official in the 200+ years of impeachments in the United States. Dumb and unconstitutional, IMO. Ironically I would oppose impeaching Biden if such a clause were in effect -- I don't even want to risk Heels Up Harris being "acting President" for a few weeks while the Senate puts sleepy Joe on trial. No doubt she'd be in a vengeful mood and issue some godawful "Executive Orders" during her reign as "acting President". Bleh!
Some patriotic Texas Republican needs to move to scrap this stupid "temporary suspension" clause. Innocent UNTIL proven guilty!
That clause is written into the Texas State Constitution, so it would require a Constitutional amendment.