Sorry, I should have been more clear. The MWO to change to the flip safety was issued over a year before StdPro's rebuild contract, so barring some very unusual circumstance, the safety would not be a push type.
The misalignment of the recoil plate is easy enough to check, but my concern would be a bent or twisted stock or a broken out plug around the recoil plate screw that has been improperly glued back in. There should be very little play between recoil plate and receiver tang - almost none - that's why it seemed odd.
Several really talented stock repair folks around here if my guess turns out to be correct. A few years ago I bought an 03A3 with a badly shattered stock. With nothing more than basic skills with wood, I got it fixed and shooting straight - and learned an awful lot about how all the innards worked together. Hope yours is a simpler fix, but even thinking through the process will help you gain a better understanding.