Johnathan,
The chamber insert will move the bullet approximately 1/4 inch further away from the rifling, causing the bullet to have as extra free "jump" before engaging the rifling. This type of bullet "jump" is usually, but not always, detrimental to accuracy. This "jump" also called "freebore" is built into Weatherby rifles to allow greater velocity withour overpressure.
The .308, or 7.62X51 NATO by its' military designation, shoots exactly the same bullet diameter and weights as the .30-06. In fact, the .308 cartridge case is merely a .30-06 case shortened and blown out a bit.
The .308 round will cycle the action perfectly. It will not, however, be as reliable in feeding as the .30-06, due to the shorter overall cartridge length. If you wish to have a .308 M-1 Garand, your best bet would be do do what I did and have your Garand rebarrelled with a match grade .308 barrel. This option will give you far better accuracy than an insert.
Whle I am quite pleased with my .308 Garand, if I had it to do over again, I think that I would have left it in .30-06, enjoyed the old warhorse as a plinker, and relied on other, more modern rifles for my super accurate .308 shooting.