I've just gotta get this off my chest. I've been a fairly serious cyclist since the early 80s. I guess I’m part of the "lycra crowd" when I'm doing a long road ride and sometimes when I'm MTBing as well, but I'm not so far gone that I shave my legs or anything like that. My best result in competition came in the TX state downhill championship, in 01 I think it was, where I won the "old guy" class (LOL). At any rate I keep reading on this forum about "slowing down" and "bikes weren't made for going that fast" and other such BS. I haven't even built a motorized bicycle yet and I exceed 50mph on both my road and MTB about 6 times a week and whadaya know, I'm still here to talk about it. Now does that mean I want to dump it at that speed? Of course not. But I also use quality components on my bikes and good safety gear. It seems like many in the motorized bicycle world build their bikes with spit and shoe strings and expect it to hold up. This goes hand in hand with the number of pics and videos that I've seen of MBers riding with no safety gear what so ever. Jeez, at the very least wear a good quality helmet. Any quality bicycle is more than capable of exceeding 30mph on flat ground with a strong rider, even my fat azz can do that for short periods. To expect less from a motor powered bike is just dumb. With that in mind, one's component and safety gear selection should become clear. The idea that exceeding a particular speed is going to lead to Armageddon can only be based on poor rider/builder choices and/or rider skill. These should be taken into account whenever someone is going to throw their proverbial leg over any 2 wheeled vehicle.
OK, I’m off of my soapbox now. So on to some possible solutions; #1 you’re running a lot of mass in your tires that probably isn’t needed with good shielded tires. #2 anytime you have a flat, analyze the tube to see where exactly the puncture is in relation to the rim and the tire so that the cause can be pinpointed. In your post you stated that it looked like the valve stem had been cut by the rim. This should only happen if the metal around the valve stem hole is rough and/or sharp and not covered by the rim strip, so look and feel to see if that is the case. If it is, use fine sand paper or a file to smooth it out or at least make sure that the rim strip is covering any sharp edges. While running 60psi in the tire, I wouldn’t expect it to slip or shift any significant amount so you might try this; when mounting the tire on the rim, inflate it to it's max psi maybe even a little more and check around the wheel to make sure that the tire’s bead is seated on the rim. Once you’ve made sure of that, deflate it. As it deflates you’ll hear and maybe even see the tube shift into a more natural position inside the tire. This is because when we cram it in there and inflate it, it may not be in the best position for it, but it can’t get there because of the pressure. Once again make sure that the stem is centered in the valve stem hole as you inflate it to your riding pressure. This is something I do every time I change a tube, hope it helps.
PS here is a great site to look at reviews of just about any MTB component, in this case tires.