The rocking motion is not normal and not wanted. That may be why your barrel was touching the stock. You have already worked on it, so I suppose you intend to keep it. May as well determine where the mid-receiver is grounding out and shave some off until you get the pillars solid.
I have a student who purchased a 700 with the same Hogue overmolded stock.....I'm not a fan. His also was not free floated.
I think maybe Remington, with the latest X-Mark Pro adjustable trigger, is trying to compete with the Savage Accu-Trigger. Neither the Accu-Trigger or the X-Mark Pro fulfill my needs and are mediocre at best in my opinion. Over the years I have adjusted more than few older Remington and Savage factory trigger with good success, although both are a pain to get the desired trigger pull weight, avoid the over-travel and keep the engagement within safety guidelines. That probably is why Timney and others have been so successful, they make a better product and have less liability risk.
As you point out the general rule for longer bullets like the 175 grain need more twist rate, so the 1 in 10 would appear to be better with the longer bullets, than the 1 in 12 of Remington barrels in the past. Not always, but that is the general rule.
I think your issues with the stock and trigger should make those A-Max 168 perform better, but if you can get better, or acceptable results out of the 175-178-180 grain range, you will probably have better results on those longer distance shots. Is your gun the 20" threaded barrel length? Are you chorongraphing loads as you proceed? What are the bullet speed results you are getting?