Like LoneWolf, I find that my .30 calibers all like bullets in the 165-168 grain weights. Almost all .30 cal barrels these days are a 1:10 twist and that seems to be the weight they like to stabilize.
My Handi in .308 shoots into two inches at 100 yards. Like you, I've been frustrated at times, trying to get it to make some really tight groups. I'm relatively new to the Handi addiction and just learning all the pecadillos of these little rifles, but the main thing to remember is that they're hunting rifles. By that, I mean they're not benchrest rifles. While some of them shoot extremely well, the vast majority of them provide yeoman service to people who simply need a good, solid, inexpensive rifle.
My go-to load for .308 is any good 165-168 bullet over 43.0 grains of Reloder 15 with a standard primer, either CCI 200 or WLR. This load shoots extremely well in just about any .308 rifle I've tried it in, turning in well under MOA in most rifles, and it shoots monotonously well in my Handi, grouping between 1.5 and 2.0 inches. Cold barrel or hot, dirty or clean, they all fall in the same two inches.
Understand, I'm not trying to downplay your desire for tiny groups. We all strive for tiny groups. However, I'm becoming convinced that tiny groups aren't the be-all and end-all for a hunting rifle. The benchrest guys and the custom rifle makers have taught us a lot about accurate rifles and we've all benefited from their craft. However, I'm not sure that lovely, tiny groups are important for a hunting rifle.
The range where I like to shoot has a metal gong at the 200 yard line. It's simply a piece of steel, one foot square. It's a lot of fun to ding that piece of steel. And fairly easy with a good rifle. After shooting that gong for a while, I've become convinced to try and make some steel targets, simple 9" disks, to hang at the 300 yard line. The heart-lung area of a standard whitetail deer is about 9" in diameter. It would seem to me that if I can hit a 9" disk at whatever yardage, every time, then the rifle is doing what it is supposed to be doing.
Will I keep trying to find that magical, one-hole load? You betcha! But, I'm not going to obsess over it like I have in the past.