It definitely has the range. The headwinds going west right now are brutal. Most JetBlue transcons (sans LAX/SFO since those are on A321s) are making tech stops in DEN/ABQ/LAS or other cities for refueling.

Converseley, eastbound transcons are benefiting from gnarly tailwinds and getting in crazy early.

Usually the pilots have enough time to make an announcement, but not always. Typically a diversion occurs after the plane has been in a holding pattern for at least a few minutes. The pilots calculate how much longer they can hold with their current fuel, talk to dispatch, decide when they'll divert and whether it will be to their filed alternate or another airport based on current weather and other factors. Before the diversion decision is made they will usually be able to check the weather at the alternate, find out which runway is in use, and pull out the charts for that airport. When they tell ATC that they're going to be diverting, they change their course to the alternate, start the descent, brief the arrival, inform the company so the airport will be expecting them. If they were holding at a high enough altitude, they'll have a few minutes to get this all done before descending below 10,000'. (I don't know JetBlue's policies, but most airlines have a policy of not making PAs below 10,000'.) After all of this, they'll inform the FAs and make a PA to the passengers.

Sometimes there isn't a holding period preceding the diversion. And occasionally the airplane is already at a low altitude (11,000' or so) when the diversion occurs. So the pilots have to get all of this work done pretty quickly. If they're below 10,000' by the time they're prepared for the approach they might not have time to make a PA. I agree that whenever possible the pilots should make diversion announcements, but if it's a choice between briefing the approach or making a PA I'd prefer they do the former.