Basically this. If you have a mount with 2 “channels”, top and bottom, and you can get a screw in both the top and bottom on a single stud that mount is never coming off the wall. Then you just use snap toggles near the edge of the mount to prevent it from flexing or tilting slightly. Like you said, TVs weigh nothing these days anyways.

If you are gutting the bathroom remove the subfloor, sister the joists with level joists, then reinstall the subfloor.

If you are not gutting leveling compound or mortar and cement board is the way to go. A skilled tile man can nearly level a floor with varying thinset thickness.

Edit - I’ll echo the “try and identify the cause of sagging first” comments. I have a 100 year old house and things have naturally moved around a bit, but it’s mostly just dips between the joists and things being out of plumb. An entire room dipping so much is pretty sus and it could just happen again after you fix it if the problem is ongoing.

You can change the direction of the vent on a dryer if it helps. You should be able to see the circles on the side of the machine you can punch out to do so. I'm not sure about venting down though, don't see how that connection could be made when putting the dryer in place.

You could also do a stacked setup if that gives you more vent options, the dryer is always on top.

Possible water got into a junction box in the ceiling below and is causing a minor short? When you have a bad connection, for whatever reason, it’s common devices “downstream” from the fault to lose all power.

There is a type of lift called a "straddle stacker" that lifts without counterweight by having legs out front. Looks like this is doing the same with hydraulic legs when it lifts/lowers, but those can come up after the load is on the lift because it's centered. Pretty neat idea looks like it would be very stable on troublesome terrain.