I’d ride Cut Gate out and back in preference to Doctor’s Gate every time tbh. You can reach the top of Doctor’s Gate off road if you’re prepared to, say, ride the footpath that heads up from the Monk’s Road junction on Chunal to Mill Hill then along the slabbed section of the Pennine Way to Snake Summit, though you’d have to churn up Chunal on the road to reach the start point. And you may have issues with walkers / your conscience depending on how early you are.
As an erstwhile Glossop local, I’ve always thought of Doctor’s Gate as a nice run and a scenic, but not very flowy bike trail, but the main problem is simply reaching the top of it. To be fair, early morning, the slog up the A57 from Glossop is actually okay. The road that side is quite open and the gradient’s pretty reasonable, so you could just pump up your tyres and do that if you really want to ride down the thing.