From everything I can gather, canards offer significant gains in performance characteristics, and most the drawbacks apply only if you remove the tail. Why not mix them together? What does Piaggio know that we don't?
Canards are not used in general aviation simply because they have complexe interaction with the main wing which can leeds to issues with stability and being at the front of the CG their effect on flight dynamics may be highly unintuitive in some situations. Therefore you need electronic control to adapt the pilot input to the control surface in order for the plane to fly as expected, and that's why in my opinion it was not suited to GA where electronic was expensive until the past few years. But they are plenty of jet fighter using them as they have the ability to greatly increase performance and maneuvrability.
In addition 3 surface aircraft means more wetted aera thus more drag, complex design and control, more mechanical moving parts which will often drive the price up.
You can find more information looking at the patent issued in 1988 by Piaggio here : https://worldwide.espacenet.com/publicationDetails/description?CC=US&NR=4746081A&KC=A&FT=D&ND=&date=19880524&DB=&locale=en_EP
canards offer significant gains in performance characteristics
The benefit of a canard wing on the Avanti is to shift the main wing back so the main spar does not run through the cabin. Note that the canard does not have flaps, so it avoids some of the drawbacks of canard wings. Also, its span is small and its free vortices hit the engine gondolas, so they have no chance to mess up the spanwise angle of attack distribution over the wing (as most other canards do).
For pitch control and for trimming the airplane with flaps down, the Avanti has a regular T-tail, so the canard's task really is restricted to shifting the wing back. This is a clever design but its use is restricted to small business airplanes with straight wings. Airliners and most swept wing airplanes in general would not benefit from the small backward shift a similar canard wing could offer, so their conventional layout has merits. In most GA aircraft, the spar runs below the pilot seat and can well be integrated with the general layout, so again an added canard wing would not improve the design.
The first aeroplane ever to fly under its own power, the Wright Flyer, was a canard design. Once they better understood the issues of stability and control, the Wrights experimented with a three-surface arrangement before abandoning the foreplane altogether after the French pattern. Admittedly nobody yet quite knew what they were doing, but the Wrights' abandonment is symptomatic of the problems the canard surface faces.
As has been pointed out, the Piaggio Avanti adopted a foreplane primarily to shift the main spar aft. Another solution to that problem is to sweep the main wing forward, as was done on the HFB Hansajet. Both types are/were medium-size bizjets with the same problem - how to max the cabin size when a main spar keeps getting in the way. Forward sweep has been adopted by sailplanes on occasion for much the same reason, but a three-surface design would impair the airflow over the main wing.
Another successful three-surface family comprises the several variants and derivatives of the Sukhoi Su-27 jet fighter which added a canard, such as the Su-30 and Su-33, but here the goal is supermanoeuvrability.
I am less jaundiced over canards than some who have posted answers here - some designs have been outstandingly successful, although the last one in general aviation that I can recall off hand was the pioneer-era ASL Valkyrie. But I would say that they are a lot harder to get right than a tail, and only worth pursuing in exceptional circumstances - whether you keep the tail or not.