The 223 is a much better cartridge than the 22-250.
The 22-250 is SAAMI registered at 65kpsi and has a practical limit for handloads at 67kpsi.
The 223 is SAAMI registered at 55 kpsi and has a practical limit for handloads at 75 kpsi
With light bullets, the fastest a 22-250 can push the bullet slows down the fastest a 223 can push that bullet with in a few feet.
The 22-250 has big advantage with heavy bullets, but due to some bullets spinning apart, factory barrels for 22-250 have slow twists, and can only stabilize light bullets.
But if you already have wasted money on the 22-250 reamer and have a take off 223 barrel you want to screw around with, and the action lying around is a Mauser bolt face, then the way to get the reamer pilot to reach the bore is with a series of steps with drills and a boring bar.
Or you could hack saw off some of the old barrel, if you have enough shank to re thread.