Disc squeal is not caused by brakes that are not centered or the shape of the leading or trailing edges. It is caused by disc contamination. This contamination could be foreign material of sorts as mentioned or it could be an uneven deposition of disc pad material onto the disc. Friction doesn't work as most people think in that two objects cleanly slide across each other. If they slide, no matter what the two surfaces are, the one lays down a layer of material onto the other. Think of a car that skids on a tarmac road. The tyre lays down a layer of rubber on the road. This interaction is not one-way but transient. Should the car tyre run on a circular track like say a road bike's brake pad on a rim, the action is transient. The rubber migrates to the rim and back. This is called adhesive friction and is mostly the type of friction you find on a brake disc. The disc never operates under purely adhesive friction and some abrasive friction is also present, depending on whether it is an organic or metallic disc pad on there. Nevertheless, the system demands an even layer of brake pad material deposited onto the disc so that the two surfaces can slide cleanly over each other. The two surfaces comprise the same material and the transient, adherent friction action is smooth and even. Should this layer become disturbed, spotty or uneven, the two surfaces start to slide unevenly in a stick/slip fashion at a frequency that's audible. The famous squeal. This means the layer of brake pad material on the disc is uneven.
The only way to get this right is to remove it but as you know, it is a pretty tough material (it is brake pad) and will not just dissolve with a bit of brake cleaner, which was really just designed to remove black brake dust from car wheels. It has to be abraded off - sanded off. It is a tough job and best done with the disc off the bike and on a belt sander. The idea is to have a finished product with a good even satin finish. It is such a lot of elbow grease (even with a belt sander) that manufacturers simply advocate a replacement. New disc requires new, uncontaminated pads and a very careful bedding in. The OP said he bedded his brakes in by riding a couple of hundred miles. This is wrong and almost certainly the cause of the problem.
Bedding in requires about 20 slow-downs (not stops) per brake, until brake force is at its maximum. This means you have to cycle up to speed, slow down by braking moderately and release the brake just before the bike comes to a stop. Do NOT stop and hold the brake down. This way too much material will be deposited onto the disc and you will have a pad-shaped high spot on the disc. Do this for each brake individually so that you can control the process. Don't attempt to brake with both levers during this process.
Don't swap pad types without completely cleaning the disc first by sanding it thoroughly. (I.e. organic to metal and vice versa).
Don't skip the process.
Don't expect that your mechanic knows this.
Don't kill your careful bedding in labour by an incorrect procedure after a long hard run with brakes. At the end of the run at the bottom of the hill, don't stop and hold the brakes in. If they are very hot, this will again cause an uneven deposition of brake pad material onto the disc.
Disc brakes are complex and problematic and I expect huge consumer dissatisfaction once they become the mainstream brakes for all bicycles. I don't see enough education at mechanic level on the issue and not even the big expensive mechanic schools have this in the curriculum.
