This is what I do. I can give you my oppinion as a handyman.
You can do as suggested and go over the carpet. In my oppinion this is just lazy, but doable. The problems with doing it this was is the lack of a snug fit, floor to ceiling. Also if there is ever any kind of water damage, or even a drink spilled near that wall. The carpet under the studs will absord and retain that water, resulting in mildew.
You do NOT have to remove the entire carpet.The carpet can be cut away to fit the framing. This is not difficult, and with the drywall and baseboards, you would never notice the cut. New tack strip along the new wall, and it would look good.
I would get some other estimates. Find someone willing to do the extra work.
either way that carpet is going to have to be dealt with. Either now or in a few years when you do replace it. It will be a bigger pain, to try and take up the carpet with a wall on top of it IMHO. It is annoying.