Can a Remington 700 in .223 safely shoot 5.56mm?
Yes, the rifle I carried for LE sniper was a Remington 700 Varmint in 223. I used M193 Mil Ball (5.56) in it exclusively until I retired and had furnish my own ammo (reloads).
I couldn't begin to guess how many thousand s of rounds I thought through this gun.
I also shot a ton of M193 guard bullets out of my Colt SP1 223 AR.
Of course that was before the Internet told me I couldn't do it.
I certainly would go through the expense of re-barreling.