Before getting into the Answer proper I want to say that brushing lacquer is not an easy finish to use, any reassurances by a manufacturer notwithstanding :-) It's even more difficult to use well, i.e. to achieve a really nice result, and this goes double for large surfaces. So without realising it you set yourself a difficult task from the outset and the product's own instructions let you down (more on this below) so don't be too hard on yourself that you ran into difficulties.
even going over a new coat with thinner to make it really wet
This was the only outright mistake I think you made. By applying lacquer thinner to the previous, dry, surface you softened it thereby ensuring partially-dissolved lacquer would be picked up by the brush loaded with fresh lacquer.
This is the sort of thing that inevitably leads to texture problems1.
Reading through the technical sheet for the product certain things stand out to me in relation to the issues you experienced.
Apply using a good quality bristle brush or applicator.
Seems you're doing exactly as recommended here but I personally wonder if china bristle (AKA hog bristle) is actually the ideal brush for this product. Hog hair is a coarse bristle and in general these are suited to slow-drying finishes that leave plenty of time for brush marks or other texture to settle out2.
If any coat has dried more than 6 hours, lightly sand with 220 grit sandpaper before recoating to avoid brush marks.
You mention sanding with 420 paper, which is fine for sanding between coats but a bit fine for taking down fairly prominent texture like you've been getting (the idea being, IF you sand between coats, that you get the surface good and flat before the next coat goes on so you don't risk compounding the problem).
One thing I note is conspicuous in its absence is the lack of a caution not to overwork the surface (they only mention it in passing in relation to preventing bubbles in the finish). Not overworking the surface is absolutely cornerstone advice with any finish capable of redissolving itself — for best results it's pretty much vital that you lay the finish on in long, smooth strokes and do as much as you can to never go back over an area twice. If you spot a bleb, an errant hair from the brush or something else you fight the urge to deal with it now and just leave it to sort out after drying.
I've read that when people are using shellac for french finishes they move on to sanding with progressively finer sand paper with a final round of polishing. Is that what people usually do after reaching their desired number of coats or is this just totally misapplied?
Yes and this is exactly what I was just about to advise. (Note this isn't traditional French polishing, that involves rubbing on shellac in many very thin coats using a tightly bundled pad to achieve the final ultra-smooth surface.)
What to do now
I would abandon the idea of getting a lovely surface directly from the product for starters. It's possible that you can do this but it is difficult to do well and there's no guarantee that final coat will dry perfectly, so I think it's safest not to try at this stage.
And now after six coats you should have applied plenty of lacquer for sanding flat and buffing/polishing so you're well set up for this next process.
There are a few previous Q&As that cover the basics but for a full article on the subject see this extract on Google Books from Fine Woodworking's More Finishes and Finishing Techniques, Rubbing Out a Finish by Jeff Jewitt.
1 As a rule finishes of all sorts work best over well-dried preceding coats and this is especially true of finishes such as shellac and lacquer which can readily dissolve their dried state.
2 According to their technical data Watco's brushing lacquer does shrink by a factor of five or more during drying, which could achieve the same end as a long drying time which allows settling, but I'd still personally want less texture in each coat than you can get from a brush like this.