They do! Its called Chronobreak. They have used it many times in professional matches successfully, but it doesn't work in 100% of scenarios.
This is not unusual. Bugs happen and when there's a lot of money on the line you need a rollback system.
Interesting. I don't think I've seen this in any other competitive game, which is why I'm surprised. But maybe I'm just lacking information
In StarCraft II you can resume from replay (and maybe in previous games as well), but I don't know if this has ever actually been used due to bugs in the game engine during competitive play.
I implemented a debug replay system for an American Football game when I worked at Sony. The AI hierarchy and all animation and physics state was recorded and game could be rewound to figure out why an NPC made a poor decision. Only needed to track state for the duration of a single play. This was on PS2 which I think had 32MB of memory but the dev systems had an extra 96MB which we could use for this sort of thing.
Hell. Replay review in American Football and other sports is a form of rewinding and fixing bugs in officiating.
So it's certainly done in real life when money's on the line.