I do not currently own the 223 UV, I did have one for a couple years and when i found the right load, it shot little clusters at 100yds, very accurate. I did notice quite a variation, however, on loads it didn't like, and I really have never had the issues many speak about as far as resting on the receiver, etc. I hold and rest my rifles just under the forearm screw. That's how I hunt and shoot, and if my rifles do not conform to MY style, they go down the road.
I also have a savage 11FL. This is not a stevens, but my understanding is the stevens is simply the older version of Savage. I have one savage over 30years old and another over 20. The 11 FL is about 3 yrs old. While the accu trigger is very nice, I cannot say it was much better than the trigger I had on the UV.
One big difference I noticed right off the bat, was the savage was very accurate with a greater variety of loads. POI might change, some, but group size was always 1.5 or so. The loads it liked are usually under 1".
While the handi is a true switch barrel/ caliber rifle, it can also be accomplished in the savage/stevens line. Sometime very simple, sometimes not so simple, but never as easy as the Handi is.
I have one that started life as a 30-06, not now has a 338-06 A Square barrel residing on it. I have a 243, that has stayed that way ,and the 11FL that started as 223 and now is also a 243.
Here is a pic of how it groups Federal power Shok 100gr factory ammo. Group size is 0.179" for 3 shots. This is this rifles pet load.

The UV did just aswell with it's pet load, which was H335 behind a Hornady 40gr Vmax bullet.
Good luck on your choice.