As aqwert said, gymnasts use a spring floor.
As a metric for how high a gymnast can launch, I checked out this video of a quadruple twist in a floor routine. Kenzo Shirai is 1.64m (5'5") tall. At 1:01, his center of gravity is, say, 1.5 times his height, giving us 2.46m -- .01m higher than the men's world record high jump. Again, this is with a springboard.
Obviously, unless serious modifications are made to the gymnast's form, this would be very hard to perform with precision such that the bar is not knocked over. Getting your timing right over the bar is hard enough with the Fosbury flop.
But this wasn't your main question: why would the officials strip the liberty of choosing a technique from the athletes?
First: it may be dangerous. It's hard to tell, since there hasn't been much experimentation with the method, as it's already been shown to be inferior. Two-footed jumping is usually what complete novices do intuitively on their first few jumps, and it's definitely unwieldy.
Second: it's inferior, so why not just mandate the tried-and-true, biomechanically superior technique? For the same reason the Amateur Trapshooting Association implicitly bans the use of anything other than a shotgun, there is no logical reason to voluntarily handicap yourself, and if nobody is going to, you might as well make it illegal to further structure the event.
In case the springboard argument wasn't convincing enough, consider why basketball players never take off with two feet to perform a layup. The sharply angled position you get yourself in during takeoff allows one to quickly convert horizontal momentum to vertical, since the leg is locked. This would rely almost entirely on muscular strength with two feet, since you could not be at a sideways angle without one of your legs bending, resulting in either less horizontal momentum (a slower approach), an inefficient horizontal -> vertical conversion, and less vertical momentum as a result of the two.
Source: ex-NCAA high jumper