That's a great question & maybe wordy answer, but I will try. Have a look at the picture I included in post#4 & the related description of how a test bar can be used. I attached picture of mine. Depending on your lathe you can use an MT sleeve adapter or just just buy the appropriate MT taper ended test bar. My spindle is MT5 so I use the MT3/MT5 adapter which I have confirmed is accurate. This allows me to use the MT3 test bar for subsequent tail stock work.
Pre-qualifiers: this relates to lathes of the type with removable HS where HS deviation is possible. This also assumes a newish machine with straight ways. The beds have not eroded away a hollow close to chuck. If that's the case, all bets are off, setup can only be a tradeoff compromise relative to chuck distance. Its an additional problem. And this discussion has zero to do with tailstock alignment. The TS must not be integrated into this initial test.
Personally, I think its best to get the bed as reasonably close to begin with even if it means somewhat undoing a prior setup. Why? because this is the very same datum surface that our DTI will ride along via the carriage when measuring the test bar pointing out into space completely independent of the ways. So as best you can, confirm the bed is not bending upward or downward relative to HS when viewed from side. And also is not twisting CW or CCW relative to HS when viewed from end. Unfortunately ($) to do this right is best accomplished with a precision level.
The test bar locates snugly into the HS MT socket and the extended portion exaggerates the HS axis relative to the 'mean' initial bed axis. Traverse an indicator down the length of the bar, both along the horizontal plane & vertical plane. We are attempting to discern if HS spindle axis is pointing inward/outward viewed from top and/or upward/downward viewed from the side relative to ways. If significant axis deviation is noted and dominated by the HS being mechanically out of alignment, that issue needs to be resolved first. If its a small deviation where we are satisfied HS alignment is as close as we will ever get, then we have arrived at the point of what I'll call the where classic methods like RDM come into play. The remainder of work is addressing lathe bed twist and ideally concluding with real cutting conditions. BTW if you doubt the significance of HS deviation as a source contribution, do the simple trig calculation for yourself. Consider a measly 0.001" over the length of HS block & extrapolate to the end of test bar. It translates into a significant amount of taper that actually represents a lot of equivalent lathe bed twist to accomplish. And something as measly as 0.001" HS rotation is nothing on a set screw, mere degrees of screw rotation depending on the thread pitch.
Here is my own summary. Others may have a different perspective. Taper cutting is a function of 2 somewhat independent sources. It could be 100% HS related, or it could be 100% lathe bed deviation related, or it could be some % blend of both. Unfortunately the sources could serve to counter/mask one another or they also could cascade & exaggerate one another. Adjustment may well be an iterative thing, but hopefully by standardizing one deviation source (ideally the one with lesser +/- limits) in order to conform the more dominant or significant source, we can proceed logically & efficiently & not chase our tails too much.