There's an extensive write-up on the 30/06-308 chamber insert in Jerry Kunhausen's book The U.S. M1/M14 rifles.
The Navy wanted some 308 Garands, but couldn't get funding to buy new barrels, but research funds were unlimited. So, they spent several million of our dollars on the chamber insert project, which was a failure. (Oh,well, it's just taxpayer money).
There were two methods of inserts.
One was developed by FN, and was a heat/freeze friction fit method.
The U.S. method involved grooves in the chamber, with corresponding ridges on the insert. This was press fit, and then locked in by firing a number of over-pressure proof loads to expand the insert.
Neither method worked well, and lacked good accuracy. There were also issues with freebore, and the difference in the 308 vs 30/06 chamber and headspace.
The civilian-available inserts are a BAD idea, since even the two methods used above would often extract with a round. Firing a 308 in a 30/06 chamber isn't a real good idea. Since you would not see the insert ejected from the M1, this entire exersize is somewhat like Russian Roulette.