The idea sounds faintly troubling, in that the aether has become some magical place where you can just send unwanted energy and everything will be ok, but so long as it doesn't allow FTL or form a privileged reference frame then you might just about be ok, from the point of view of not annihilating physics as we know it.
(also, make sure you have a read of Project Rho's entry on reactionless drives, because this does apply to you here even if they seem reactionless, because the knock-on effects will be similar. Note Burnside's Advice: Friends Don't Let Friends Use Reactionless Drives In Their Universes.)
Big problem number one, then, is not so much "physics goes foom", but "no planet in your universe is ever safe ever again", because you're allowing super-efficient high-thrust low-power propulsion and enhanced heat sinking, and that means that anyone with access to aether drive technology can make stealth relativistic kinetic kill vehicles on a whim and hit anything with enough energy to totally smoosh it, and you might not even be able to see the projectil being accelerated in the first place so it will be a total surprise (unlike laser-driven or antimatter-driven projectiles, which are deeply unsubtle for years ahead of time). You might have to handwave maximum velocities, but they'll have to be fast enough for convenient space travel which implies that kinetic weapons with tens or hundred of km/s muzzle velocity could be commonplace and straightfoward. That's hazardous.
Big problem number two: is the aether "at rest"? If so, what it is "rest" relative to? If the aether is at "rest" relative to the sun, what will happen as the sun progresses in its orbit? We're moving at about 230km/s around the centre of the galaxy. If the aether rotates with the galaxy, that might be ok, but if it doesn't gravitationally interact, there's a problem there (like how that even happened in the first place). What happens in half a galactic year's time? Will turning on an aether drive suddenly fire you out of the solar system at 460km/s? If you're not willing to wait that long, what about if you fly to Barnard's Star, will activating your drive give you a big 90km/s boost? If the aether is at rest relative to something else, turning on your drive wil subject you to massive accelerations, suitable for interstellar travel but not interplanetary work so much. Also, it risks becoming a massive special frame of reference, and that has some Potentially Bad knock-on effects vis-a-vis relativity (alternatively, it might form the basis of a non-time-travelling FTL mechanism, depending on which way you wish to wave your hands). Given this, you might consider some other kind of reactionless drive that doesn't accidentally create a privileged reference frame, like an Alcubierre warp or perhaps something more like a tachyon rocket (with the latter perhaps being modified into a less physics-destructive aether drive, with a bit of thought? would certainly make a change from the usual warp drives people often have)
With that out of the way, here's something else to think about: with an aether drive, I'm imparting some momentum to whatever it is the aether is made up of. Presumably it has its own equivalent of mass, or you have other problems (like, what speed does an aetheric particle have after you've interacted with it?). These particles don't interact with each other or the rest of the universe, so they'll never shed any energy, and you probably won't get any equivalent of thermal blooming. These potentially high energy particles can only interact with other aether drives. Suddenly I can interact with stuff at a distance without all that tedious mucking about with gravity or electromagnetism!
On an object with significant mass of its own (like a moon or planet) I can create a drive beam that can impart momentum to any active aether drive it can hit. This can be used to accelerate objects away from the planet, decelerate incoming objects and, interestingly, repel anyone in range with an active aether drive. Ergo, you may need permission from local traffic control if you want to be able to fly in a straight line. It also means you might not be able to use aether drives to do terminal guidance of missiles or manoevering of space fighters (because point defense can push you away).
You could therefore also presumably use it to beam energy and information through any solid object (including neutron stars and the like) and provide remote power and communications. Point-to-point communications without line of sight or the need for satellites seems useful. I'm not quite sure how power generation would work, but you could presumably make something as dumb as an aetheric turbine (shoot an aether beam at one side of a wheel covered in aether drives) so I'm sure something much more sensible could be made. This potentially allows spacecraft to be remotely powered and propelled and have a limited heat signature. Maybe not enough to be a sort of submarine in space, but hey: they're your hands to wave.
Finally (for now) this is probably a better way of driving a starship than any other. Use an aether beam to push em, and a magnetic sail to brake em. All you need to worry about then is shielding, but you have the semi-realistic prospect of relativistic spaceflight ahead of you.