I think the implied problem is that its location makes it a better choice than JFK or Newark for most travelers, but if all the airlines shifted their flights there for passenger convenience, it couldn't handle the flights because there aren't enough runways and there's no room to add more. The traffic in and out of the area might be the maximum volume for the airspace, but only because the airlines are flying flights into JFK and Newark that they'd prefer to fly into LGA if it could handle it (and if it got a major upgrade).
In principle, this seems like a good idea. It would make the ATCs' jobs simpler to have to manage only two approaches instead of three, and it would have the added advantage of keeping the flights farther away from the skyscrapers for safety reasons.
With that said, they should also tie it to Newark. Ideally, it should be an express train with three stops: JFK, Grand Central, and Newark. They could possibly share the Amtrak route from Newark to Penn Station, adding a new stretch underground that turns northward right before Penn Station (so that trains don't fly through there at speeds approaching 100 MPH) and up to Grand Central), then rejoin the line. Then eliminate the largely redundant LIRR line that parallels the subway in the stretch from Hunters Point to Sutphin Blvd so that you can run at high speeds all the way from Grand Central almost all the way to the airport, and finally, share the short stretch of AirTrain track from Sutphin to JFK.
For a great example of the way JFK and Newark should work, fly into or out of Heathrow and take the Heathrow Express to London Paddington... except underground, obviously. Both airports are only about 10 miles out. A rail line running at the speed of the Heathrow Express would mean about a seven minute trip to Grand Central Station from either airport, versus up to a 45 minute cab ride from JFK to midtown. You'd basically run trains in a 30 minute loop back and forth.
Obviously you'd have to share the tracks with Amtrak, so you might have some slowdowns while their trains are running, but otherwise, it would be a really effective way to reuse some existing lines. And then you could ditch LGA entirely and turn it into... whatever.