The idea is that different equipment is better suited for different individuals. By mandating a specific kind of equipment for all athletes, you'd be benefiting some individuals over others.
For example, road bikes have different frame shapes that are suitable depending on your torso length compared to leg length.
Of course, the natural next step would be allow some flexibility (different frame size but same material), but you can see how that could be a slippery slope of legislation and lobbying that would end up in a similar situation to where we are today?
You -should- be able to just mandate Hubs and cranks to get around this.
Or make sponsors provide a 'slew' of wheels and cranksets for all their riders (i.e. spares as well) that can get inspected before the race, and are randomly distributed between the riders sponsored.
Buuuuut UCI hasn't even figured out doping, so lol
Suggesting everyone should have to use a specific bike is exactly a suggestion like that - bikes have sizes and shapes that are suited to riders of particular body sizes and shapes.
For example, road bikes have different frame shapes that are suitable depending on your torso length compared to leg length.
Of course, the natural next step would be allow some flexibility (different frame size but same material), but you can see how that could be a slippery slope of legislation and lobbying that would end up in a similar situation to where we are today?