Why wouldn't KJFK offer the ILS for 22L?
JFK ARR ILS 22L/22R DEP 22R


  If conditions dictate the use of JFK ILS RWY 22L approach, then (1) the JFK Area owns Belmont airspace 3,000' and below, (2) releases Coney airspace to LGA and LIB, (3) JFK cannot depart 31L, (4) ILS 22R unavailable if LGA departs RWY 13.


LGA ILS 22, DEP 13


  LGA Coney, Maspeth, and Whitestone climbs are available.


LGA ILS 22, DEP 13 (No Belmont)


  If conditions dictate the use of LGA ILS RWY 22 while departing RWY 13, then LGA departures are restricted to Maspeth and Coney climbs only*.


So, based on those two things, if LGA doesn't own the Belmont airspace, then Whitestone Climbs (WC) are disallowed. Therefore, when JFK uses VOR 22L, it allows LGA to use the Whitestone Climb off of runway 13, which is preferred for noise abatement.

*Personal note: this application was created prior to the addition of GLDMN, NTHNS, TNNIS RNAV SIDs which are now used instead of Maspeth and Coney.

Sources: 


Previously based in LGA at a regional airline
NY Airspace Application 
http://tfmlearning.faa.gov/NY_Airspace/NY_Airspace_Pkg/NY_Airspace.swf
(Only works in Firefox)