To preface, I'm a microbiologist, so I have skin in the science game. I hate how these articles often have science illiterate authors or authors who are imprecise with their wording. They repeat misinformation on basic topics that science educators have been striving to correct for decades, perpetuating the cycle.
...the study shows once again how evolution throws up multiple solutions to basic problems...
In this case, it's the "mysterious force of evolution that whips up solutions to problems". Evolution doesn't create solutions. There is no guiding force behind evolution.
Evolution through natural selection selects for existing solutions that were generated randomly through mutation, increasing the frequency of that trait because those without either die or are outcompeted. What happens if a trait is required for survival but no organisms have it? They all die. That's why over 99% of all multicellular species that have ever lived on Earth are extinct. If you include microbes, make that 99.99999%.
Is there any evidence that species with higher rates of mutation, and therefore higher chances of evolutionary advantages, stick around? Or is rate of mutation not something which can be encoded in DNA / biology?
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.