How is it possible for supersonic aircraft to push air out of the way?
The idea that force can only be transferred/propagated at the speed of sound is a simplification.  It's a very reasonable simplification, which works in a very wide array of cases, but it's a simplification.

A fundamental concept in gas mechanics is the "mean free path length."  This is how far a molecule can travel on average before it collides with another.  This is on the order of hundreds of nanometers for pressures you'd see at reasonable altitudes (68 nm at STP).  If you are looking at processes that occur at scales much larger than this, it is very reasonable to model their behaviors as transmitting things at the speed of sound, because these statistical effects lead to a very natural expected speed of propagation.  It can be seen as a result of the central limit theorem applied to a very large number of molecular collisions where the drift velocity (average movement of particles) is relatively smooth.

As you push things, this nice model breaks down.  As you approach the sound barrier, you get pressure gradients that get shorter and shorter.  Near the speed of sound, these gradients start to get into the single-digit-multiples of this mean path length.  At this point, we can no longer assume that we can just model everything with the nice clean statistical models we had earlier.

What actually ends up happening in most situations is that you get a "shock."  Far from the flying body, the air transmits force at the speed of sound with respect to the unmoving air (or slowly moving, if there's a wind).  Right up against the flying body, the air transmits force at the speed of sound with respect to the flying body.  At "reasonable" speeds, there's a smooth shift from one to the other, where the pressure gradients aren't so steep that we can't call them pressure gradients.

At supersonic speeds, that smooths shift doesn't happen.  Instead, we end up with a thin shock region (on the order of 1 μm or thinner) where it's not reasonable to model the air as having a "pressure."  It acts more like a bunch of billiard balls being flung along.  This region, a few mean free path lengths wide, is where the usual rules we're used to, break down enough to resolve the physics.  On each side of the shock, "normal" speed-of-sound behaviors occur.  Inside the shock, it's messy.