Don't ask more of your rifle then it was designed to do.
I too have rifles with very light contour barrels---#1 contours 22 inch in length and .515 at the muzzle.
They are hunting rifles and all I ask of them are good 3 shot groups ---shots fired one after the other and then a good cool down before the next group.
It is IMO unrealistic to ask such a light barreled rifle to shoot 5 or 10 shot groups one after the other.
Shot in the manner that I have described, my rifles will put 3 shots well under 1 inch---just fine for a hunting rifle.
If you want to shoot more rapidly, get something with a good quality heavy barrel and you will be able to extend your number of shots.
There are also some calibers that lend themselves to this better than others--that is, some calibers will heat a barrel up faster than others.
Dipper