As one of the best wood finishing authors puts it, "A wax finish is the closest thing to no finish at all." It provides minimal protection, especially to water. In my wood finishing class, I always point on there is a difference between a wax finish and a waxed finish. The latter is a film-forming finish that you've polished with a wax top coat.

You have not applied "three coats" of Briwax. Wax contains a lot of solvents to make it soft and softens the lower layers. While you have made three applications of wax, there is one layer of wax, not three distinguishable layers. And the way to apply wax is to buff off as much as you can, then do a little more. You want just a few molecules of wax thick or you will have a soft, gummy/waxy finish that will attract dirt. IMO, wax finishes are only good for decorative items such as picture frames or carvings; things that rarely get touched and never have anything set on them.

Paint thinner is a solvent for wax, so you will succeed in partially dissolving what's there and smearing it around. You could really do the same by just applying another application of wax and buffing most of it off.

Wax is quite difficult to remove completely. Lots of clean cloths or paper towels and solvent such as paint thinner will get most, but not all, of it off. So simply "putting on some poly" is not likely to stick as poly does not adhere well to wax.

Using steel wool between coats or before coats of finish is also a bad idea as you will never remove the steel shards and bury them in a finish where they might later rust. If you need to use anything use non-woven abrasive such as Scotch-brite.

Finally, 90% of the things out there called "Tung Oil [Finish]" are not oil, and do not contain any tung oil at all. They are typically thinned varnish (e.g., Formby's) or (linseed)oil-varnish blends (e.g., Minwax). While they may be perfectly good products, they are not in any way tung oil. Most tung oil products will say 100% tung oil and will not contain any "petroleum distillates). Tung oil (the real stuff) has lots of disadvantages:
- expensive
- visual results virtually identical to much cheaper linseed oil
- if you do not apply correctly, you will get uncorrectable spotting and the only cure is to strip and refinish
- provides only slightly more protection than wax
- needs to be periodically renewed

So my recommendations would be either of these:
- go ahead with your plan to regularly re-wax and consider this just maintenance for your chosen finish

- strip and refinish with a non-poly (alkyd or phenolic) varnish.

Here is a link that might be useful: lies, damned lies, and tung oil finishes