That asteroid diagram is a bit misleading. It's showing the distribution of the asteroids, but it greatly exaggerates the sizes of all the bodies relative to the sizes of the orbits. Even the dot representing the Sun is too big. If the body sizes were to scale in that diagram, all of the planets & asteroids would be smaller than a pixel, so they'd be invisible.
The asteroid belt isn't dense. A typical estimate of the mean distance between asteroids is around 1,000,000 km (2.6× the distance between the Earth and the Moon), but that's for asteroids of diameter 1 km and larger. There are probably estimates for smaller bodies, but they aren't easy to find, although there's probably useful info in the articles linked at What is the average distance between objects in our asteroid belt?
There are around 10,000 asteroids with a diameter of 10 km or greater, but they aren't close to each other, so your "chain reaction" scenario with a comet launching a fleet of Mount Everest sized asteroids heading towards the Earth isn't really possible. Even with much smaller asteroids it's still extremely unlikely.
From Wikipedia, here's a diagram show the numbers of asteroids of different sizes.
Also from that article,
The orbital distribution of the asteroids reaches a maximum at an eccentricity of around 0.07 and an inclination below 4°. Thus although a typical asteroid has a relatively circular orbit and lies near the plane of the ecliptic, some asteroid orbits can be highly eccentric or travel well outside the ecliptic plane.
Most of the asteroids of any significant size are in well-behaved orbits of low eccentricity. Major collisions aren't impossible, but they have low probability. And that includes collisions with bodies passing through the asteroid belt, like comets or spacecraft.
According to veteran asteroid hunter Tom Gehrels of the University of Arizona, in this Scientific American article from 1997,
In some ways, the asteroid belt is actually emptier than we might like. In the early 1990s, the National Aeronautics and Space Administration wanted the Galileo spacecraft to encounter an asteroid while it was passing through the asteroid belt on its way to Jupiter. But it took some effort to find an object that was located even roughly along Galileo's path. Special targeting was required to reach this object, but the result was the first close-up view of an asteroid, the one called Gaspra.
The number of objects in the asteroid belt increases steeply with decreasing size, but even at micrometer sizes the Pioneer spacecraft were hit only a few times during their passage.
It takes a lot of energy to perturb the orbit of a large asteroid. Yes, a large comet could supply that energy, and send dangerous material towards us. We do need to keep watch on the asteroids, but we don't expect a fleet of them to head our way anytime soon. Most of the exciting planetary bombardments happened billions of years ago, when the young Solar System still had plenty of bodies with "wild" eccentric orbits. The last big episode, known as the Late Heavy Bombardment, happened around 4 billion years ago... or perhaps it didn't actually happen. ;)
The Late Heavy Bombardment (LHB), or lunar cataclysm, is a hypothesized event thought to have occurred approximately 4.1 to 3.8 billion years (Ga) ago, at a time corresponding to the Neohadean and Eoarchean eras on Earth. According to the hypothesis, during this interval, a disproportionately large number of asteroids collided with the early terrestrial planets in the inner Solar System, including Mercury, Venus, Earth and Mars.
Since 2018, the existence of the Late Heavy Bombardment has been questioned.