Some backlash is inevitable, and is one of the things to take into account when machining. Years ago, I spent a bit of time reworking my Atlas crossfeed parts to get my backlash of 0.009"-0.010" down to 0.002"-0.003". Any less and I would experience binding. It's now back up to 0.004"-0.005".
So, when doing diameter reducing out->in cuts, run the compound inward to a good stopping point to get its threads pre-staged to handle the pressing forces.
Opposite when doing boring in->out cuts, prestage the compound by turning its screw backwards.
And, all crossfeed readings are done with its leadscrew loaded in the cutward direction.
Aligning the spindle and tailstock's centers' points is a quick and easy way to align the tailstock, and is just fine for most general-purpose work. But, for precision machining on shafts, your earlier dual-diameters comparison is the preferred way.
I have a straight/polished 1/2" x 12" SS steel bar that I've centerdrilled true, and use it as a gauging bar between centers. Mounted, I simply run my toolpost mounted dialgauge up/down the rod to determine tailstock centering, and to setup offsets for tapers...