I’ve had a buddy come up hunting one year, before we went out on day 1 we went to the range to confirm zero. He’s an experienced shooter and groups were all over the place, he noticed his rear ring was loose and he re torqued it and all seemed well. Fast forward a few days later after driving around in his SxS and hiking, he shoots a 3x3 wt buck around 75-80yds. point of aim was a broadside heart/lungs shot, it drops like a stone. When we got up to it it was hit in the spine straight above the vitals, his rear ring was loose again. So it definitely contributed to accuracy issues, mm’s of play at the gun turn into feet the farther out you get.
I can completely confirm a loose ring/mount entirely shifting my POI from my POA specifically on a Savage Axis rifle.
Savage Axis XP .308 Win Left Hand - Factory Trigger - MDT LSS First Gen Left Hand Chassis - Witt Clamp On Brake - Magwedge 20MOA Picatinny Rail - Nikon Bushmaster 2 4-12 in one picture but swapped it for a Bushnell Banner 2 6-18 Scope - Ergo Grip Tactical Deluxe Palm Shelf Grip - Caldwell Bipod - Magpul ACS Stock - Howa Slip On Bolt Knob Cover.
Proper torque specs and leaving loc-tite to cure for a full 24 hours is a must on top of what you said how ATV's and SxS's shake everything loose real well.
Here is one of my 100 yard targets I was using to get dialed in, the rear of the scope base started to come loose on me and you can see the point of aim to point of impact shift very evident in the upper right corner.
Norinco .308/7.62x51 surplus ammunition shoots great out of my rifle, the grouping just above the tape measure consist of 5 shots and the shifting group was 10 shots.
The included Bushnell 3x9 with a Savage Axis combo leaves a lot to desire but I will say the one I took off my .223 Axis back in 2014 is still holding up just fire with over 15K rounds fired through it with the majority being 7.62x39, .223 and .308 Win.
I bought my .223 Axis combo back at Wholesale sports in 2014? for just over $300 as it was the display model everyone put their dirty mitts all over and that rifle will shoot the same hole all day no matter how hot you get the barrel.
Another pig but a lot less lipstick, Savage Axis .223 Rem Right Hand - Factory stock - Factory Trigger - Harris Bipod - CTKPrecision #MPOD1 - Magwedge 0 MAO Rail - Redfield Revolution 3-9x Scope.
I do similarly I've been having lots of friends and family drop off their new package rifles to show me and I always pull the scope, rings, and bases and pull the barreled action out of the stock. Degrease and oil up all the metalwork, reinstall the action and torque to spec. Degrease the fasteners and re-install the bases and rings with correct torque. Luckily my basement is big enough to laser bore sight to 30 feet, and that gets me within an inch or two at 25 yards.
I find if I'm 2 inches low on the laser then it prints pretty close to that at 25, and close to dead-on at 100 which helps keep the ammo costs down when sighting in (because they never seen to bring ammo with them).
I've always had at least okay accuracy out of the Axis package rifles. Usually no diamonds in the rough but always "usable" accuracy. The scopes are junk optically and the tracking is pretty whacky but I've never had one not zero, and I've always been able to hand the rifle back to someone with a target and say "if you missed you either dropped the rifle or suck at shooting".
You have a very solid course of action for mounting your optics and I can confirm DEGREASE EVERYTHING is a must have when getting things installed properly.
The factory Bushnell optic does leave a lot to desire but in the same sentence if you damage/destroy that optic its not a huge loss, I'm still using the Bushnell 3x9 that came with my .223 Axis in 2014 and luckily it has held up to my neglect over the years.