Weatherby rifles are accurate, but the group size will generally degrade after the third or fourth shot, if the particular rifle has a thin-tube barrel. This is common to any brand of rifle. This is no problem for a hunting rifle, where only the first one or two shots matter. What's important there is that the first shot always go to the same place as the last time you shot the rifle...
If you want small groups from a five- or ten-shot string, a heavy or heavy-ish barrel is needed. (A slow rate of fire is always helpful, of course.)
If you plan to shoot at long range, heavy bullets do better than lighter ones. Heavy (and thus longer) bullets require a faster rate of twist to provide stability.
My only experience with a Weatherby is my Mark V with a #2 profile barrel (26", medium weight) in .30-'06. I rebedded the forearm and installed a Canjar trigger. It has always been capable of five-shot, one MOA groups; three-shot groups of 7/8 MOA or less are common, and there is the occasional three-shot, 1/2 MOA group. Ol' Pet is now about 33 years old, with some 4,000 +/- rounds through it.
I suggest you give some more thought to your "I want", and research the .223 cartridge a bit more. There have been many discussions about bullet weight and behavior, here; all available via the "search" function.
Art