Nowadays, sure, but keep in mind that the "US" didn't extend beyond the east coast when the Puritans first settled here. You might be able to make an argument that there's no cultural influence from the colonial days that lasted until today (although I'd disagree with that sentiment), but otherwise, where would you expect any cultural influence in the rest of the US to have come from?

(To be clear, I'm not saying that there weren't existing cultures there before the US expanded out further west, but I imagine most people would agree that the US today isn't culturally as influenced by them as much as from the the colonies and pre-expansion US.)

New England is perceived as less religious than the South, but one reason for that is that New England's moral perceptions had a strong influence on US political beliefs. In other words, the Puritans morphed into the Congregationalists who morphed into the Unitarians, who basically took over (in the 19th Century) US political thinking (or at least the Left side of it), giving the appearance that New England does not having any particular or special moral or religious beliefs (at least to those on the Left half of the US political divide).

I grew up in New England and have lived in the South and in California, and IMHO morality is a bigger determinant of the behavior of the average person in New England than it is in the other places I've lived (all in the US). The South and California are more pragmatic, less moralistic.

That's a good point. When something is within the usual for someone's experience, it's not going to be as obvious, so it becomes the baseline that's used to compare other things to. For stuff like religion, it's easy to assume that your amount is normal, and having more (if you don't feel like you have much) or less (if you do feel like you have a lot) is unusual.

I don't have any experience living outside of the northeast (although not New England specifically since high school), but I definitely agree that there's certainly more religion in New England than might be obvious from the outside (more Catholic than the rest of the country, which also might explain some of the differences).