Great question! In fact, some of the most successful pathogens (Herpes viruses are a excellent examples) have evolved with precisely this strategy. Herpes viruses have very low virulence, but infect nearly every human on the planet, and stick around for life.
But there's an evolutionary trade-off to this strategy. Herpes viruses have giant genomes (for a virus), because they need a ton of genes that counteract the immune system and perform other feats of cellular control so that they can stay dormant for long periods of time. While dormant (the technical term is "latent"), the virus is not transmissible, so while it persists for a long period of time, they actually don't transmit very often.
Other viruses (influenza is a good counter example), are readily transmissible throughout most of their life cycle, but end up being cleared in a few weeks by most people. And yet influenza also spreads around the globe each year, so this also seems to be a pretty effective strategy.
It's really hard to ask questions like "why doesn't influenza behave like herpes?" When it comes to evolution, there are many successful strategies. Might as well ask, "why don't fish have feathers?" "Why don't sea birds have gills so that they can stay under water?" "Why aren't humans single-celled organisms?"