A few points to add to the excellent points already mentioned:
1. Modern transports are now have almost neutral static stability, so there is little "downforce" provided from the horizontal stabilizer as it is. Even on convential airliners there is less downforce than you might expect, with the horizontal stab countering the pitching moment on the wing but the CG does not necessarily have to be ahead of the CP.
2. In high speed aircraft the requirement of thin airfoils without high lift augmentation creates the possibility of leading edge stall. The high pitch rate that results from such a stall combined with the canard being well ahead of the CG means that the canard AoA will continue to be high a lot longer due to the pitch rate than would otherwise be the case. In fact, the pitch rate can lead to the AoA actually increasing as the nose descends. It is possible that the nose could then even "tuck under", leading to a roll about the lateral (y) axis. Passengers haven't learned to appreciate that sort of maneuver for some reason.
3. Canards are very useful in reducing pitch transients, so you can get faster reaction to pitch changes. Great for a fighter, not all that useful for transport aircraft.
4. As others have stated, the airflow behind a canard is downwash. However, because the canard will have less span than the wing, the effect is a lower AoA on the root section of the wing and a higher AoA towards the tips. This creates two problems, in that it increases induced drag due to the spanwise distribution of lift coefficient (why beyond the scope of this discussion), and also the obvious adverse effect of increasing the likelyhood of a tip stall on the wings -- not a good thing.
This can be offset with forward sweep, but that also results in new problems in structure and a lot "harder" ride in turbulence, among other things.
In balance, don't hold your breath!