Once you get over the speed of sound, the aerodynamics don't change so drastically as they do in the subsonic-to-supersonic transition (which is what crashed the early attempts to exceed mach 1). In that transition, the center of lift on a wing surface goes from about 1/4 of the way back from the leading edge to the middle of the surface, and that tends to make the craft nose-down. Once the pressures on the surfaces are determined by shock wave patterns, it is more an issue of what angles those shocks have and where they might hit another surface, but you still need lift in about the same place at supersonic and hypersonic speeds. And, of course, the planes need to start and end their flights as subsonic aircraft.
Modern fighter jets are designed to be neutrally stable instead of positive stable in flight, so they can maneuver more flexibly. That comes at the expense of needing a computer to act faster than a human pilot can react to keep then from going awry. Modern jets can climb without pointing the nose up, first, for instance. I expect that computer needs to be faster and smarter for aircraft that get into the "hyper"sonic range.
But, to me, the biggest issue is getting jet engines to operate at those speeds, and also at low enough speeds to take off and land the thing. Supersonic combustion ramjets (SCRAMJETS) are good to have at hypersonic velocity, but not what you want for landing at subsonic velocities.
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