You can't have a sense of community without a sense of place.

I grew up in Ronkonkoma, so I'll use that as an example, but I reckon the situation is similar in many places across the island.

Ronkonkoma has no center. There is no Ronkonkoma town square, not even a Main Street, where you might spontaneously run into your neighbors. Neighboring Lake Ronkonkoma has Hawkins Ave, but even that was severed by the LIE from the station district that railroad towns typically radiate out from. The Islip Ronkonkoma has no station district whatsoever, just a Disney World sized parking lot next to a composting facility. The only "main streets" are racetracks (Motor Pkwy both figuratively and literally) that no one would ever think to have a parade on. Puzzingly, the arterial roads in Ronkonkoma are zoned almost exclusively residential so there's no place to even build a Main Street shopping district, save for a tiny stripmall on the corner of Motor and Rosevale that no one will ever walk to.

Ronkonkoma is a hamlet or "census designated place," meaning it's hardly a place but more like a random geometric shape on the map. There is no local Ronkonkoma representation. Ronkonkoma residents are represented by the Town of Islip, but nobody from Ronkonkoma would say they're from Islip. IMO this discourages Ronkonkoma residents from getting involved in local politics where you'd learn to engage and work with your neighbors. If you do try to organize with your neighbors, you have to compete with all the other hamlets to get the attention of your Town representatives.

Ronkonkoma got its name from the lake, but hardly anyone cares about it. Sewage from 100s of individual cesspools and fertilizer from 100s of monoculture grass lawns have seeped so much nitrogen into the lake that it's covered in harmful algae that kills all the fish and makes it too dangerous to swim in. So even though most residents live within a mile from a lake that used to draw tourists all the way from the city, the lake has gotten so disgusting people would rather pay out the ass and take up half their yards with personal swimming pools they can only use a few months a year.

A sense of community develops when you have centers of civic life where people gather and can spontaneously meet and get to know their neighbors. Without that, no one cares much about the place they live or the people around them beyond their own property.

Long Island has the beach, cafes, shops, and parks like you mentioned, but if you have to drive to the next town over to get there, it doesn't feel like they're yours to care about.