You should be able to do cobra for the short term after leaving your current position so you don't have to have any uncovered time. Any reasonable employer understands this. If they don't, they aren't worth working for anyway.

I once had a recruiter tell me much the same thing. I responded that by being an asshat and applying pressure that is totally unreasonable demonstrates that he cares more about his placement bonus. This will brand them as a recruiter that doesn't care much about working with people which will only damage his reputation as somebody who is difficult to work with and not other developers will want to work with you. Then reminded them that relationship between a recruiter and a potential employee should be one of mutual respect and if they aren't capable of that I will look to the next recruiter (who btw has the same list of jobs you do).

In general it seems that many of the jobs around get listed with multiple recruiting companies. I have a good relationship with a few of the recruiters (who tend to change jobs more frequently than most people change their underwear). This makes it easy for me to turn away the bad ones and get some really focused help from the recruiters when I need them.