The fastest way to leave the Solar System is to pass by as many of the Gas Giants as you can, and use their gravity to slingshot you faster. As there are no planets outside of the ecliptic, it would not be advantageous to avoid the ecliptic plane. This holds true until we start to get really fast space probes.
Furthermore, passing by the planets would give you a chance to abort, by slingshoting around them to return to Earth, in the event of a quick catastrophic failure (Entirely possible).
Just to prove this point, New Horizon's left Earth as the fastest spacecraft to ever leave Earth Orbit, reaching the orbit of the Moon in only 9 hours. Yet it will never catch up to the Voyager probes, because they used the gravity of both Jupiter and Saturn to speed up. I can't put it any better than Wikipedia, so here goes:
New Horizons is often given the title of Fastest Spacecraft Ever
Launched, although the Helios probes are arguably the holders of that
title as a result of speed gained while falling toward the Sun. New
Horizons, however, achieved the highest launch velocity and thus left
Earth faster than any other spacecraft to date. It is also the first
spacecraft launched directly into a solar escape trajectory, which
requires an approximate velocity of 16.5 km/s (59,000 km/h; 37,000
mph), plus losses, all to be provided by the launcher. However, it
will not be the fastest spacecraft to leave the Solar System. This
record is held by Voyager 1, currently traveling at 17.145 km/s
(61,720 km/h; 38,350 mph) relative to the Sun. Voyager 1 attained
greater hyperbolic excess velocity from Jupiter and Saturn
gravitational slingshots than New Horizons.
Furthermore, the amount of objects in the ecliptic plane is vastly overstated. Space is really really big, and we have to plan very carefully to get a spacecraft to another planet purposely. Even a small miss will cause issues. There isn't much there, and there is much to gain by going through the ecliptic plane.
The day that we have spacecraft capable of very high thrust for extended periods of time, we likely won't need this shortcut, but for now, it's an invaluable tool.