There's no reason I can think of why this won't work.
A couple of things to watch - make sure the cutting edge trails the direction of rotation so that it doesn't dig in, and thoroughly protect the lathe bearings and bed from abrasive particles. A careful clean-up after a sharpening session would be wise, too.
I've used a disc on a lathe for decades - plane irons, chisels, turning tools, scissors, kitchen knives, shears - all sorts. I used water soluble PVA - glue straight to the metal and soak the paper off when you've finished
Be meticulous with your wood dust removal prior to doing it on a wood lathe and don't do it last thing before shutting up the workshop. Grinding sparks and fine dust have a habit of taking a few minutes to make their combination felt. Likewise using a non dedicated Disc or Belt sander.
I suppose you have to be aware that the farther towards the circumference of the disc the faster material will be removed and therefore compensate for this when sharpening plane irons.
I had a go with the scary sharp system, and very good it is indeed. Especially if you don't have a sink in your workshop because there's no need for stone flattening.
But the down side is that it tears if you so much as look at it the wrong way. I'd be concerned that whizzing around on a lathe, and without an ultra flat substrate, the specialist scary sharp papers would be shredded in a flash.
I don't think the OP meant scary sharp as in fine paper, I think it was just the principle of sharpening on flat abrasive paper. The question is different to the heading - I think if the reshaping of a turning gouge on 10000 grit (or whatever it is) paper would take a while.
Yeah I meant sandpaper sorry for being vague. I usually use about 3 grades, on glass, with good results. The lathe would be more for initial shaping, but it might work for the polish too? Looking forward to giving it a shot!