To clarify a bit, it sounds like. Trying to imagine it to come up with a solution.
( ) (your heating system)
| === | (baseboards/sheetrock/ductwork that presumably connects to something)
______ (platform that needs to be replaced)
===== (some sort of pipe/duct)
______ (concrete immovable slab)
With all that in there, he probably gave you a “I don’t want to mess with it” bid to account for the fact that if me breaks any of that pipe/ductwork he has to replace it and pay for the damage it will cause.
If I were you, I’d look specifically where the duct work meets the next connector. If it’s insulated, you might have to pull the insulation off to find out. It got in there in pieces, so you might be able to remove it in pieces easily. Barring that, I’d take measurements and head to the hardware store to see what kind of connections are available, then probably cut it out with the hopes of putting a piece back in with connectors on each side, then reinsulating. (sawzalls are magic, powerful magic that must be respected)
The baseboards can be pulled off usually with a crowbar. Depending on how successful you are, you may not need to buy new ones.
The sheetrock, I would score with a utility knife and break off at a point above where I will be working (if possible). Maybe 18 inches or two feet above the floor? Basically score it, hammer a hole, and then pull it off in the biggest chunks you can manage. You don’t want to cut it directly with a saw because of the studs/pipes/electrical that are probably behind it. Once you’re done you can cut a sheet to fit and “patch” the bottom 1–2’ of those walls along the line you scored, so spend some time making sure it breaks along that line.
So that leaves the thing that was leaking and sits on the floor you want to replace. I think that really depends on how heavy, how mobile it is while connected, etc. The right way to do it is no doubt to disconnect it, get it out of there, replace the floor, and then put it back in. Sometimes with a friend or a jack, you can raise it, hold it with a crossbeam or something to the walls (you’ll have exposed studs if you remove the sheetrock) and then remove your old floor, slide and position your new floor, and get it all connected.
That’s the thing though, you run the same risk the professional did of breaking something unintended and it ends up costing much more than expected. It’s also a lot of work in a closet that you presumably don’t really have enough room to move around like you’d like to.
Interested to know how it goes and good luck.