Post some of your current targets and tell us more about your loading and load development practices, components, rest/bags, and optics. We need to know how close to the potential of a .223 you are shooting now.
The difference in absolute precision between a .223 and 6BR at 100-200 yards might matter to a competitive shooter (though no winning BR shooters are using either of these cartridges at 100-200 yards) where a few thousandths of an inch in group size can make the difference between first place and tenth, but that doesn't sound like you.
With my CZ527 Varmint which was a 1:9 twist, the best I was able to do with it was about .5-.6" 3-shot groups and .75-.8" 5-shot groups. That is averages, not cherry picking my best. I know that there is more potential with that caliber. I sold that rifle to a friend and he noticed an issue with the crown, that possibly could've been hindering it's grouping potential. I've been able to do about the same with my White Oak barreled AR15.
That was using 60gr FBHP Sierras and 69gr SMKs, usually under H335 or Varget, respectively.
My bench setup is a Caldwell Rock BR front rest with protektor front and rear bags. Scope was a 20x Nitrex TR1, and recently a Sightron SIII 10-50x60.... which will be what I put on the Savage I get next.
I used Lapua brass, FL sized and trimmed before firing forming, then neck sized only until the shoulder needed pushed back. Lee collet neck sizer. Seated the bullets to about a .015" jump with the 69gr SMKs, which required the use of a single shot adapter. I used a Wilson seater die with a K&M arbor press, which I could also measure neck tension while seating the bullets. I sometimes would sort by bullet weight but rarely noticed a difference.
As far as load development, I never really played around with the seating depth much, but I did do a ladder test at 200yds and found that with the 69gr SMK and Varget, it liked just about a full compressed load the best, which I believe was around 26grs or just under?
I know that I still have a lot to learn and may not see the benefits of a 6BR compared to the .223, yet. But Im thinking that if a 6BR is in my future, why wait?