Oh no, rate of mutation is definitely a thing and is controlled by several factors. A big one is generation time, which is what it sounds like, the time between each generation. The copying of DNA is a source of mutations. This is why many controlled experiments on evolution are done with bacteria, who have super low generation times. For example, depending on temperature, the generation of many salmonella species is around 20-30 minutes. That lets you crank out massive numbers of potential mutations, then introduce a selective pressure, like an antibiotic the species normally isn't resistant to or an energy source it normally can't utilize, and see what happens.

To answer your question, yes, a higher mutation rate would confer an advantage. To a point. Most mutations are deleterious and usually lead to death, a few are benign and do nothing (at that point), and a very rare few are immediately advantageous. As long as the rate of mutation isn't so high that the deleterious mutations combined with whatever other pressures are wiping out the population, more mutation means more chances to have the right trait to deal with a novel pressure or, very rarely, do something better.

I grew up believing the same until somewhere in high school, when I started taking science seriously.

... a jumble of local maximums and chance.

I really like how you phrased this. I'm totally stealing it.