Hey tictac, I've got a similar situation on one of my Remington rifles. It is a S&S 24" 308Win M700BDL-DM. I'd mounted a BIG Leupold on it with their Dual Dovetail mounts and couldn't get on the paper with scope adjustment. It was shooting to the right.
I looked around a bit and had a set of the Redfield style (Adjustable Rear) bases and put them on it. I've had to move the rear of the scope to the "Left" a bit and I can notice it when I look down on the rifle and scope, but it doesn't hurt a thing.
My particular Remington rifle seems to be very picky about which bullet it likes the best, which is extremely unusual. But, the big news is that I think the majority of "my rifles problem" is related to the "magical" barrel harmonics with that thin, long barrel on it. Now that I've got the groups somewhat centered, some brands/styles of bullets still go to the right of the Point of Aim and some to the left.
But, the more I shoot it, the closer the separate groups seem to get. Makes no sense at all to me. Nothing is wrong with the bedding, or trigger, or scope, so I first suspected it was just due to a lack of "trigger-time" on my part.
I do not know why your crosshair is not centered when the "adjustments" are centered. I would have expected the crosshair to be centered when the adjustment screws were centered too.
Anyway, I'd recommend you center your crosshairs in the scope and then use the "Lateral Adjustment" of your rear base to get the Point of Impact as close to center as possible. Then fine tune with the scope adjustments. And, don't worry about that rear base adjustment " IF " you are getting nice tight round groups.
Good hunting and clean 1-shot kills, Hot Core