My close tolerance boring experience is confined to model engine liners in various kinds of steel & cast iron. The target has to be within 0.0005" but more importantly, of a specific finish at that final bore. That's not exactly an easy combo to hit. I have a super stiff boring bar that has a shank diameter probably 70% of the bore ID. I tried HSS but preferred sharp inserts for consistency. Part of that was likely my inability to dress & set up HSS properly & maybe cheapo import HSS boring bar. They just aren't as common anymore.
What I think you will discover if you get into lapping after turning is bore gages are measuring an average of whatever hill tops they are residing on. That may be fine for your purpose, in fact maybe even desirable if you are using retainer type adhesive. Lapping its quite revealing. Your beautiful boring bar finish at 0.0005" tolerance inevitably turns into 'micro-stripes' where the lapping has flattened the hill tops with a matt finish but the troughs are still shiny. My pics don't do a great job of showing this properly but I'd recommend you try yourself. This is why I was recommending you change up your traverse feed selection on final spring passes. If you don't, you are just spring passing the bottom of the vee. But that has little to do with the dimension across the crests (think of it as threading which is essentially what is happening). Yes a more rounded tool tip minimizes this effect, but I feel also at resisting DOC so even more need for spring pass. Lapping also reveals other effects like ever so slight bore difference between the opening of the hole & closer to the chuck. Lapping (if done properly) doesn't lie, its visually quite apparent until the whole surface is entirely matt & now removing native material.
Of course all this is very material + cutting tool + machine rigidity dependant. The recommendation to stabilize temperature prior to measurement is very important at these tolerance levels. Re measurement, I like snap gauges too but I would recommend a good quality bore gage set against a known standard. Not only can you get good consistent direct readings without in-between-device chances of error but you can repeat this many times & at different positions down the bore, 'mapping it' so to speak.
If you can get to within 0.0001" using lathe tools & no grinding/lapping secondary finishing, you are a better man than me. The way I think about it, if every commercial 'cylinder bore making' company could hit the appropriate tolerance /finish with cutting tools alone & mitigate grinding / lapping / honing operations, why would they go through all that added expense?