Yes, the new heavier barrels will shoot better. They do not string anywhere near as much as the thin barrels do. You should be able to get 1.5 inch groups with good ammo with the heavier barrels.
The thin barrels can be made to shoot much better with the addition of a support strut. I hope to be posting pictures of my struts that I will have for sale within a few days. They work just as well as the commercial one's at less than half the price. They are pretty darn good looking also.
If your looking at buying a Mini, a new one would give you good groups without any work.
If you find a good used thin barrel Mini for a low price, it can be made to shoot. My thin barreled mini shot a best 3 shot group of .474" and 5 shot group of .855" at 80 yards. One flyer opened up the 5 shot group and it was the last and only 5 shot group I fired that day; I ran out of ammo. I'm not sure if it was me or the weapon. The 100 YD range was full so I couldn't shoot at 100 like I would have like to.
It was a clear, windless day and the rifle was shot from a rest on a cement shooting bench.
The Mini 14 was modified with a steel bedded action, a strut, trigger job, Mason flash hider/brake combo, 3 X 9 Nikon scope and a re cut muzzle using 55 grn factory ammo. It is an older 187 model with the 1-10 twist barrel. It also had very few rounds put through it, so it was a new, older rifle.
Best Regards, John K