The problem with grass season is the other Slams. You could shorten the hardcourt season, or, in theory, move the US Open later (it's still pretty nice in NYC in mid September). But, the USO would never give up it's slot during the last week of August and over Labor Day (in the US) weekend. People are still on vacation, and school has not started for many.
Same with the AO. Miami and Indian Wells really should be leadups to the AO, which would be ideal in early March. But, the AO is held during Australian summer and the tournament and fans benefit from that fact.
The existing summer hardcourt season really isn't that long (though it seems longer). Wimbldeon finishes around July 6. The next week, there is no hardcourt event. And then, on July 14 (at least this year), Indianapolis kicks off the hardcourt season. There are "only" five weeks of events before the USO. I always think it seems longer, but it's not.
There is also the issue of grass facilities to have a legitimate grass season. Technically plenty of grass courts to be found, but are there enough different sites to hold actual ATP tourneys over 5 weeks? This assumes, that if there was a legit grass season, there would be a grass Masters, or Masters-like tourney that would need a big facility with appropriate seating.
We may wish for an extended grass season, but I've never heard the ATP even suggest that it's an issue.
I always thought that Newport should be played during the second week of the French. I know this would mean that many top players would not play the event since they made it to the second week, but it would allow players who lost early to get a jump on Wimbledon prep and I guarantee the Newport draw would be better than it currently is. Furthermore, the top players tend not to play it anyway, because it's not in preparation for anything. At least there would be a chance that a Roddick or someone else who lost early at the French would play.