You go through LEO, you just don't park there. Think direct insertion into higher-altitude orbits, in the style beginners in Kerbal Space Program do it - i.e. going straight up through LEO and only then turning sideways to gain orbital velocity.

The risk will be much lower (you'll be spending very little time in the band filled with hypervelocity bullets), but it'll also be much more expensive. You'll need a much bigger rocket.

How do you get to the outer solar system without going through the asteroid belt?

We don't worry about hitting an asteroid if going through the belt, because the density of objects is so low.

If you wanted to stay in the asteroid belt you would have many more opportunities for a collision, so avoiding asteroids becomes a lot more important.