Actually, most departures out of LGA use the Whitestone or Flushing climbs, which involve a a turn to the left, rather than right... The Flushing climb, turns right to 180 degrees first until at 2.5DME from LGA, and then makes the right turn.
You can verify this info with BigPhilNYC, as he lives in College Point, and observes this almost daily.
Also, you're JFK ILS 13L argument is flawed... If you use the ILS 13L at JFK, you are no longer able to use the Expressway visual for LGA (which is used very often), as well as the ILS RWY4 (as previously stated). Also, the planes landing on RWY22 do not always run up the Hudson. A lot of them fly over Queens, over LGA, left or right into the Bronx, and then a 180 degree turn to RWY22. I would imagine they are still fairly high up while over the ILS RWY13L path at JFK, but it still adds a complication. Also, I don't know too many people who want to have heavy aircraft flying over Midtown Manhattan, especially close to the ESB, every 2 minutes... If something were to go wrong on the approach, God forbid, you could end up having a huge mess in Midtown, as well as near the ESB. With two planes already taking down two buildings, I don't think people would be too comfortable seeing them that close to the ESB.
In short, could the ILS to RWY 13L at JFK be done? The answer is yes (and I would love them to do it, b/c that would mean those hvy planes flying right over me, and I would have a clear shot of them all the way to the RWY)... will it happen? probably not too often..
You said they were using the ILS RWY13L at JFK? Did you mean the 31L/R? If not, do you know how far away the planes start on the ILS?