Are you talking about on-ice harnesses? When a skater is practicing in a harness on-ice, yes, they do the whole jump. If they are just working on the takeoff, they don't need a harness - would just do a big single.Do skaters practice takeoffs only in a harness or is it always the whole jump?
Off-ice jumping could be actually rotating the whole jump off the ice (usually occurs with axels) or simply jumping up and rotating - i.e. skater starts out on 2 feet and uses legs and arms to do the rotations and lands on 2 feet. Or exercises that involve jumping on an object (ex: box) to try to gain power/height on jumps that you are doing on-ice. The "mjmskating" (Mitchell Johannson Method) IG has some videos of their skaters training off ice.
The idea of sneakers having less support than skates is probably true and a good point. Hopefully, the majority of elite skaters are working with trainers who can guide them through exercises safely. Unfortunately, as with any physical activity, injuries can occur. Rachel Parsons broke her ankle about 7-8 years ago doing off-ice training (and I'm thinking she wasn't training jumps) but fortunately, it happened in the off-season and she was recovered by the start of the season.
Some rinks have an off-ice harness - one of the rinks in my area does, it depends on space - which coaches can use with their skaters. Similar to an on-ice harness, it helps the skater get a feel for rotation and air position. And landing on the floor might be a softer landing than on the ice.