I completely agree, I don't think your tires are to blame or at least not 100% for your fall dude. It's easy to blame the tires because it's the only contact between you and the road (I certainly know it's usually my initial reaction - but I think thats akin with denial), but there are a lot of reasons that tires slip - weight distribution, suspension disruption, speed/traction overload (too much speed at low traction moments) - the list goes on. But you'll notice in all of those - the driver initiates the transgression.
On new tires, warming them up its just the same as warming your engine, but you've gotta thinking about the sides. In any case, on a normal spring day after a few miles your tires will be good to go. BUT because they are new you should be careful, track or not, that means slowly increasing your lean angles, not jerking around with the throttle and the brakes, you need to be extra smooth (should aim to be all the time anyway). Complete your breaking before you corners, I wouldn't trail brake on new tires, front or back unless I've already hit the low angles. When in the corner go into maintenance throttle and stay there, keeping a nice 40/60 (front/rear) weight distribution until you start your exit.
That said - I think you probably just upset your suspension at a relatively new angle to the tires and caused a loss of traction. I've definitely done new tires at the track and after a warm-up they are good to go for the first lap, as long as your smooth on throttle and brakes.