If I were in your situation, I would take the new job and move to Gig Harbor (or wherever). I'm basing this on my experience over the last year: when they announced our jobs and kids' school would be fully remote last year (in the Boston area), we relocated to our vacation home in NH for the year. We worked remotely and kids went to in-person school there. It was great -- so nice to be in the mountains, skiing before work, etc. Really didn't miss the "amenities" of Boston all that much. Loved the different pace of life up there. We thought really hard about staying, and almost pulled the trigger, but it would have been somewhat complicated for us (e.g., possibly requiring at least one of us to eventually change jobs), and our school-aged kids ended up being split on wanting to stay (one really wanted to, one really wanted to return to the Boston area). I think if our kids had been united in wanting to stay in NH we would have figured out how to make it work. Now we are back and it's fine, but I'm spending a lot of time daydreaming about the mountains.
Based on your post, you don't have any of these complications -- your kids are young enough that they are not already attached to a home school/community (as ours were), your wife is at least potentially on board (obviously you need her buy-in), and you don't have any real job-related complications other than the $$ difference (and I really think $200-250k is enough, esp. in LCOL area). So bottom line, I think you've got a good window here to make a big move like this, and I'd go for it. NYC is a great city (I lived there for 5+ years early in my career), but I think there are plenty of better places to raise a family, and those golden handcuffs are only going to get tighter if you stay.
I hope you let us know what you end up doing.