There are 3 STARs (Standard Terminal Arrival Routes) into JFK that cross over Long Island:
PARCH 3 RNAV ARR:
PAWLING 2 ARR:
ROBER 2 ARR:
None of these have particularly pronounced zigzag patterns, so what you describe is probably not coming from the waypoints of the arrival procedure.
Looking at flightradar24.com, we can currently see ...
Coming from Southern California, you'll often be filed to one of those points, and then on to SERFR for that arrival. Putting those two points on the page gives perspective of what direction one is coming in from.
It's all RNAV stuff, so is having them on the chart strictly necessary in order to navigate? No. But it gives a little bit of a picture of what ...
Some SIDs/STARs have a restriction to only turbojet or turboprop aircraft and say piston aircraft (what I suspect you meant by "GA") should use another SID/STAR, or they may have different altitude or speed restrictions (or "expect"s) for different types.
Some, like the one shown above, don't say anything about aircraft performance; this probably means that ...
Revisions are made to improve traffic flow, adjust for infrastructure(eg local airport change in number of flights or nav beacon decommissioned), simplify procedures for pilots and ATC, or address some other routing conflict or safety issue.
This is why IFR charts and terminal procedures are only valid for two months.
For example there is a victor airway ...