LEO is pretty important, first we have people there right now, and with the notable exceptions of weather and GPS, everything I take for granted in life that works because of satellites is in LEO. Not having it for 150 years would make space exponentially more expensive in every way.

The Kurzgesagt videos are well produced but aren't the greatest for accuracy in the details on a lot of topics, sadly.

Not an AE, but from my understanding of orbital mechanics, I think we usually use a LEO orbit to insert payloads into higher orbits or onto escape trajectories. Even Starship, which could be considered a next gen orbital vehicle, relies on LEO for reloading fuel before leaving Earth.

How exactly do you propose reaching higher orbits without going through LEO first?

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.