Funny you mention Burlington, I spent a few years there! Being a college town, it's uniquely walkable and laid out around that idea. I don't think the equivalent of Church Street exists anywhere in New Jersey.

Anyway, I think Maplewood/South Orange is among the closest in NJ you can get to that; I did a bit of research when initially looking for a house to buy a few years ago around New York.

Both train stations are very walkable depending on where you buy. From where I live, for example, Maplewood station is 15 minutes and South Orange station is 25. A few friends of mine from the city have also moved here, and I meet with them every week or so for coffee and whatnot in each town's downtown area. We eventually bought a car so we could interact with the rest of New Jersey, but in terms of local needs, a car is helpful but not at all required.

For me, at least, it's firing on all cylinders with your requirements, which are pretty close to mine. Had to pay a lot when I bought though, hopefully the market isn't as crazy for you now.

Another note: we vetoed Montclair not because of the prices, but because the train lines are worse. SO/Maplewood is on a line that is always really convenient. One stop further is Millburn, which you also may want to check out.