what I thought was right quickly became mis-aligned.
I'm not familiar with that lathe but from a picture it appears to have a rotating head.
I would certainly try the double-ended MT alignment tool first. If the tailstock can be loosened do that then tighten them both. The problem will then be if the tailstock doesn't remain perfectly aligned with the lathe bed as it is moved away from the headstock, a posibiity if there is any play in the construction of the tailstock both horizontally and vertically.. On one lathe I had to fasten a piece of very thin shimstock under the tailstock to make it perfect aligned vertially.
As far as the points aligning when the headstock is brought close to the headstock the MT alignment fixture will fix that for close work which may be critical, but it might not for long spindles when the tailstock is moved away. Fortunately, when turning longer wood spindles some tiny misalignment doesn't make much practical difference.
For example if the tailstock was misaligned by a small amount, turning this hoe handle would simply introduce an insignificant bit of taper:
There are other ways to check alignment using a precision steel rod but they take a lot of work and some specialized. tools.
BTW, I know a lot of people like rotating heads but I've never been tempted. I know that can make some face turning easier but I don't have any problem with a non-rotating head. If working on the inside of something and I want to see without bending over the late, I simply turn in reverse - I have some made, for example, to make undercuts on bowl rims.
In most cases there is no need to see - The PM3520b allows sliding the headstock to the end for those who want to see with the eyes instead of the fingers.It even provides for a a way to mount a lowered toolrest to turn things larger than 20". I don't do that, don't have the big-bowl gene.
JKJ