Question about using tapered reamer to repair the Morse taper in a tailstock

Hello,

My lathe's (ESM-59) tailstock is in decent shape except the taper is a bit worn.
I find there can be some play in the drill chuck and unless I am careful the drill bit can be not on center.

It uses a #1 MT (Morse taper). I was thinking of getting a #1MT high speed steel taper from McMaster-Carr to try to ream the
taper in the tailstock. I would put the reamer in the lathe's collet and then carefully bring in the tailstock to resurface it.

Does that seem like a bad idea or is it a standard thing that is done with good results?

Is there a better way to refresh the taper in the tailstock?

Thanks,

Jan