I'd bring back ties, obviously, since I'm old school. But since that's apparently a no-go for Americans, broadly, I'd at least focus on minimizing how often they happen.
Absolutely go to 10 minutes of 3v3, and if there are any other rule tweaks that can be implemented to increase decisions prior to the skill contest, let 'em rip. I don't think anyone actually enjoys the shootout, it's just a "i hate ties and this is at least a way to prevent that" (or at least I hope so, if you LIKE the shootout please consider seeing a professional).
I think I would legitimately prefer going to 2v2 and then 1v1 after the end of the 3v3 than go to the shootout. Anything that is even slightly more hockey-like is better.