See reply in Gunsmithing forum.
I used to shoot rifle competition in the Navy, and had opportunity to watch the gun mechanics accurize M1 and M14 rifles.
The barrel/gas tube "springiness" you described is not good. I would look for interference with the sling swivel nut or stud that you installed. The front end of an M14/M1A should hookly onto the stock, and the receiver/action should pull securely into the stock with the trigger guard latched. Selective fitting of parts, peening, and fibreglas resin is used to bed the action and remove all looseness in mounting the barreled action into the stock.
Try looking up "national match M1" on the web. There are a number of sites that describe in detail the accurizing of these military rifles.
HTH
John