A good quality lathe would need sub-thou tolerances on both those alignments, wouldn't it? Even tighter for a toolroom lathe?
I think ideally, yes; practically??
If your height is within a couple of thou, you are good to go. Picture this: you get the TS perfectly aligned with the spindle center, TS quill fully retracted. What if the TS bore is out of alignment (nothing that anyone can do about that after it is machined at the factory)? Now you extend the TS quill to 3/4 of its maximum (say 3”). That center could be out by 5+thou, depending how bad the bore is out of alignment. What about the quill lock - put an indicator on it and tighten it. You’d be surprised how much the TS center can move.
Big question: does it really matter? I don’t think so (at least not as much as people believe it does). Just use the side adjust to correct any taper and away you go.
Oh, another thing: where along the lathe bed are you going to zero out the TS so that it is in perfect alignment with the HS spindle? It might be perfect 20” from the HS, but due to bed twist, wear, different TS lockdown force, etc, it will be out again anywhere else. So, again, I don’t worry too much about it.
I eyeball center the TS (using the alignment mark on the back) take a cut, measure for taper, adjust the TS until there is no taper (on subsequent cuts, if required) and call it good.
Yes, I made a ”dumbbell” test bar (which I even ground) because I wanted to see if it was faster than the eyeball method above. Using the test bar gets me closer right off the bat, but the final adjustment is still via actual cutting passes. So it actually takes longer using the bar than just eyeballing it.
Just my 2c on TS alignment.