I know Splash had one (blamed it on B'rer Fox and B'rer Bear), Little Mermaid has one (blames it on Ursula), Haunted Mansion (blames "pranky spirits"), and I think PeopleMover and Spaceship Earth just says there's been a delay.
The rides you mentioned (minus PeopleMover, which stops more than in starts) stop frequently for guests with mobility challenges/wheelchairs.
Every ride's ADA* procedures are different, although the Omnimovers have designated vehicles that they will hold at the exit for someone who is using a mobility device and cannot walk a moving walkway (HM). Some also have vehicles that can accomodate a wheelchair (Buzz Lightyear SRS). Omnimovers will also stop if someone is having an issue boarding. You'll hear these stop pretty frequently as a result.
Other rides don't have as long delays for boarding someone with a mobility disability or it doesn't cause the entire ride to stop. It's a Small World, for instance, may cause a backup at the exit, but not the entire ride.
Some rides have the announcement regardless of whether the person boarding actually is slowing down the ride. For instance, RnRC makes this announcement any time someone boards in row 3 via the exit (which has a door). When I used it, it did not take additional time verses standard exit procedures, but they made an announcement anyway. Now, someone who is in a wheelchair and has to directly transfer might take extra time; in my case, it was because I was in a boot and the door made it 1000x easier to board. They did the announcement anyway. So if you hear that announcement there, odds are it's someone boarding at the exit.
Tiana's I also noted an announcement whenever I boarded even though I was taking about the same time to board as others. I suspect it's a "just in case" thing. Those logs are difficult to board ordinarily, and even more if you've got a mobility challenge. IDK if they have to stop the logs going up ramps when someone boards via the exit, but it wouldn't surprise me.
So yeah, those rides you're hearing the announcement frequently, but they're stopping frequently as well.
PeopleMover you have to be ambulatory to ride. But they likely stop it a good bit due to the moving walkway. It also is just a problem child of a ride.
*Before someone jumps in...yes, I know the ADA is the Americans with Disabilities Act and I'm referring to mobility boarding. It's just an easier term to use.