Question

Why would it not be in a microbe’s best interest to kill its host too quickly?

Answer 1

During an infection the microbes utilizes the molecular machinery of the host for its replication and spread of the infection. Hence it is in the best interest of the microbe that it does not kill the host.

Also certain microbes like viruses only survive in their host bodies thus their survival is entirely dependent upon their host survival so they generally follow a lysogenic cycle that is they integrate their genome with the host's genome in order to replicate with the host itself.