I've had a .223 Handi-rifle with the bull barrel for several years now.
Mine is very accurate. It's at least as accurate as *I* can shoot. I won a postal rifle match with it among some guys I know at another forum. I was shooting against a couple of guys who were shooting some very serious hardware - custom built match guns that both cost them over $2,000 each.
The only disadvantages I can see in the Handi-rifle are 1) it's only a single-shot, and 2) it's a 1 in 12 twist barrel, so the newer long and heavy .224 bullets that need at least a 1 in 9 aren't suitable for it.* (I shot the match with Black Hills 52 grain match-grade ammo.)
8)
Edit to add: * I just went and did a little searching, and apparently the newer .223 Handis are rifled 1 in 9, which is good (according to the twist rate/bullet weight chart on Shilen's site) for up to 70 grainers.