And there are plenty of $4s which are I will consider opening with, but would rather take a Cutpurse: Baron, for instance.
I'd be curious to see the rest of this list. I think Baron > Cutpurse, usually, but it's also pretty close to the bottom of the totem pole.
Without an exhaustive list of the 4s, I might be missing some here, but openers I'd prefer on average to Cutpurse in no particular order: Sea Hag, Remake, Salvager, Tournament, Caravan, Militia, Baron, Young Witch, Moneylender. I think Cutpurse generally beats the $4 Villages and Throne Room, but it's probably better than Feast on average.
I'd agree on most of those except for Baron, which is a lot worse than people give it credit for. Sure, it ups your chance at an early Gold, but if you get it in a hand with four Coppers you are just screwed, and that happens a lot. I'd add Smithy and probably Bishop, too. Other folks might add Envoy too but I have an irrational hatred of that card. Thing is, though, there are a lot more $4 cards out there that you're forgetting about. Some of them are really powerful but often worse than Cutpurse
as an opener: e.g. Tournament, Conspirator. Some of them are situational: e.g. Ironworks, Potion. Some don't make any sense as openers: the Villages, Throne Room, Spy. Some are okay openers but I'd rather have Cutpurse: Remodel, Island, and Baron. Actually, I do horribly with Caravan and would probably take a Cutpurse over Caravan b/c of that, too.
Probably the worst $4 I'd open with (not counting strange setups which favor normally-bad cards like Bureaucrat and Talisman) would be Navigator.
EDIT: Councilroom says I buy Cutpurse about 47 percent of the time, and do slightly better than average whether I buy it or not (1.28 Win Rate With, 1.27 Without). So, it's not an elite opener but it's pretty good, and I feel comfortable with that evaluation.