It's true that Transcare runs EMS in many of the voluntaries, especially in the Bronx. They have Bronx Lebanon, Montefiore, and Einstein to name a few. I worked for Transcare as an EMT-B, and they should be avoided like the plague. They are profit-driven corporate healthcare at it's worst. They cut corners at every opportunity, and have no respect for their employees.
You're better off with voluntaries that run their own EMS like Columbia Presbyterian or even the Catholic hospitals under St Vincent's. I've found them to be less restrictive than working for FDNY, although you are still working under FDNY supervisors, and most pay better too. I now happily work for a voluntary, though I won't say which one. I've seen people get jammed up for things they thought were said anonymously in forums.
Overall, I do have to agree that NYC isn't the best place to work EMS. The lack of both salary and respect can be frustrating at times in a city where firefighters are worshiped as America's heroes and consider Paramedics and EMTs to be "ambulance drivers". I personally know a FDNY EMT-B crew who needed a facilities break and found the ambulance bay at the ED jammed with buses and supervisors on a busy night. Rather than add to the crowd, they decided to try the firehouse around the corner. After ringing the bell and respectfully asking to use the bathroom, they were denied. The firefighter who answered the door told them that civilians weren't permitted in the firehouse, while his idiot buddies laughed in the background.
Frustrating, to say the least.