Well that's too bad, it didn't work. Nor did different tool geometries.
I guess my options are to find a piece of PVC pipe and see how easily that turns (I doubt it'd turn too easily), use a piece of PVC pipe and mould some sort of body filler/ epoxy thing (I also can't imagine body filler turning too pleasantly without chunks of it chipping out), or I could scrap the MDF and try again with plywood.
I think I might just go for plywood. I'm not so hot on the idea of using this MDF any more since, after being wetted and dried on a radiator, it probably won't stay the same shape unless I leave it for a month to stabilize. Plus I have the plywood on hand.
That said, there's probably a likelihood that i'll experience some chipping with the plywood too, what with turning it on the end grain. Although Matthias Wandel had success turning plywood for bandsaw wheels.