Why is chassis connected to earth ground but not neutral?
Firstly, note that in some parts of the world, neutral and ground are NOT connected at the house, but back at the substation (transformer) so there may be a few volts on neutral in those systems.

But to the question : Consider if you did just connect chassis to neutral.

Then what happens if part of the house wiring fails open circuit?

1) Live (Hot) fails ... appliance stops working safely.
2) Neutral fails ... appliance stops working - with the chassis live!

This is not good.

Connect the chassis to earth and what happens when a wire fails?
1) Live (Hot) ... appliance stops working safely.
2) Neutral ... appliance stops working safely. If a current path develops between live and earth (perhaps you are trying to fix it) 10 or 20ma will trip a modern breaker disconnecting the supply.
3) Earth ... Nothing happens unless something else goes wrong. The earth failure will be caught at your next scheduled safety test. Right?