I agree,....alot of very sensible posts here relating to this particular rifle, and great recommendations. Did you post the group sizes, and how many shots per group? Sorry if I missed this.
I have owned three like this PF action, still have a 270 and 7mmRM,..sold the '06 20 years ago. You are right, nothing feeds like a push feed M70.
My 7mm, has the factory pad and is heavier, and looking at your load data, that would be stout with that hard plastic black butt plate.
Do all mentioned on screw tension and bedding,try to find some "copper out",....similar to product used by British army sniping school. It's ammonia based, so make sure you hold your breath,....but can it shine up a copper fouled bore!!!!
I removed the pressure point on the 7mm, didn't even bed the action

and look at how cheap those glass kits are, but free floated, and it did tighten groups,....but also the shape of the groups. The 3 shots always seemed to be more circular, without the usual 3rd shot opening up with the pressure point. Can you see sharp lands in the throat area?
You got me to thinking of what my pet hunting load was in my through the 80's in my '06,..so I just dug out my old records.
The last load I used for deer, was a 150gr Nosler SB BT, over 59.5 grains of IMR 4350/CC!200. This load was fast and accurate in my rifle. I remember it was alot stouter to shoot, than 66 grs of IMR 4831 pushing a 175 gr Sierra BT out the 7mm.
The '06 was a lightweight by the way, but did have a tiny red rubber factory plate. If you have that hard plastic, they are nasty on 50 rds off the bench!!
