I'm not sure I'd use one of the specialty way slide products under a head stock Petertha, since there a plastic product, I think you'd start seeing them slowly cold flow over some time under the pressure of firmly bolting the head stock down. There are and what's already been mentioned special and no doubt fairly expensive epoxy's today meant for use on machine tools. With a lathes head stock, everything you can do to ensure a firm evenly supported bedding of the head stock to the bed will help eliminate sources of chatter. The tough part is guaranteeing the head stock is in fact properly aligned until that epoxy sets.
And yes your correct, the method I detailed won't 100% identify a combination of inaccuracy between both the cross slide and head stock alignment or there misalignment whatever that amount might be. What it will do is allow measurements of what the lathe currently produces under those cutting loads. Its then a process of step by step elimination. And you have to start with some known baselines of what the lathe is currently doing. If I wanted to check if the cross slide was angled towards or away from the head stock, that's also easily done. Use the longest and known to be straight edge you have that's still short enough it can be swung 360 degrees in the lathes head stock. Lightly hold that and roughly centered in say the 4 jaw independent chuck between two of the jaws, I say lightly because you don't want heavy jaw pressure distoring your straight edge, and you only need to hold it tight enough to resist the tip pressure from an indicator or have it slip. Place an indicator / magnetic base on either the cross slide or lathe bed. Zero the indicator tip on the straight edges end closest to you, rotate the spindle and straight edge by 180 degrees. Adjust the straight edge ends in either direction until both ends indicate zero / zero. At that point you know the full length of the straight edge is square to the spindles rotation in the X axis. Now use the cross slide to indicate the length of that straight edge. That will show the cross slides exact orientation to the spindle. Because of the critical nature of these measurements, I think I'd want to use at least a 10ths or metric equivalent capable DTI.
In my opinion or at least it was true for me, machine tool alignments aren't the easiest to properly visualize in all three dimensions. Its non optional to be able to do so and why I recommended reading that Schlesinger PDF enough until your grasping what it's showing. Read through it a few times over a number of months while thinking about what its trying to teach, sooner or later it will start to click and make complete sense. While it's expensive at around $100 with shipping, a very good but hard to read through because its pretty dry companion book, would be the Connelly, Machine Tool Reconditioning. Machine Tool Reconditioning for Machine Tools by Machine Tool Publications
I'm almost positive a great deal of what's within it is directly based on Schlesingers work. Even if you never rebuild any machine tool, what's within it will vastly change anyone's opinions about poor machine tool way cleanliness and just how important proper and adequate lubrication really is. That alone was more than worth the books cost for myself. Connelly address it in his book, but by far this diagram illustrates the possible 6 degrees of freedom within any one single machine tool slide. Renishaw: Six degrees of freedom (6DoF) explained
Properly understanding that is a key concept to the rest. I'd also highly recommend the Moore Tools book Foundations of Mechanical Accuracy, a PDF of it is here, https://pearl-hifi.com/06_Lit_Archi..._Tools/Foundations_of_Mechanical_Accuracy.pdf
Since this has been a real subject of interest to myself for almost 40+ years, it's also not something that I think anyone will pick up in a few days.
Yes and without question that millionths level accuracy is way outside what ESzczesniak was asking about or what any of us could even approach in our home shops. Its still in my opinion not wasted effort to educate yourself about the basics. I've used some of the principals from the information I've listed to help check and tune some of my own machines. A full check of any machine tool using Schlesinger's methods takes quite awhile, but for most of it, your only verifying exactly what you have a single time. If the machine checks out you don't have to do the more extensive checks again for multiple years. Even though my last lathe came with an industry standard test certificate I still ran the exact same checks myself to verify those numbers were real and that those checks had even been done. There's also the fact that you can't even hope to try and compensate for any machine tool inaccuracy unless you already know where they are and by how much there out. This information is also why I maintain that well over 90% of the forum posts about basic machine tool alignments are trying to be helpful, but there based on a lot of less than well understood concepts. For most of them, they recommend lathe leveling and tramming the head and vise in on a mill and that's it. Yes those are obviously correct, but are still almost the very last step after verifying the rest with any new or used machine. If far more people would go through just the information sources I've mentioned, you'd then see a whole lot less well meant but still not quite well understood machine tool alignment posts on all these forums. I'm certainly not trying to slight anyone, I'm simply stating what the facts happen to be in general. I'd also point out, that for almost all of us, using industry standards for bearing fits, thread standards and multiple other items found in Machinery's Handbook is a common and completely accepted practice, you could hardly have a home shop and not do so. Yet most at the hobby level seem to ignore what industry uses to set up and do basic alignment checks on any new machine tool they get before it's ever put into use.