I think it does. There are genes involved in controlling mutations, and different loci can have different mutation rates. Viruses would definitely have some tweaks going on that mean that they're prone to mutations at the right loci for purpose.
The structure of a virus may also mean it's prone to different type of copying errors, or errors at certain locations and not others. That's the stuff I'm talking about.
Also, think about flu on a meta level. A successful flu virus wants to be at or near the genetic average of all viable flu viruses, such that as many possible mutations lead to viable flu viruses for that host. This maximizes the chance of that particular flu virus mutating into another viable strain that outraces the immune system, whereas viruses which are at or near the edge of the viable gene pool will tend to be weeded out. So as influenza viruses cycle back and forth through the population, any specific viruses that don't have many viable options for mutating to new strains which can outsmart the previous anti-bodies won't tend to replicate as much as ones which have many possible pathways. This would then influence the "meta design" of the virus itself, optimizing it for efficient shape-shifting (measured in : how many possible mutations lead to good variants vs shitty variants).
So there are meta-selection pressures at work, too, to create viruses for which many possible mutations lead to viable routes as possible.
EDIT: another point is about search space size. Natural selection is about optimizing a search space, so you get a fitness landscape, and the genome converges on the local maximum of the fitness function. Successfully evading defenses requires two things: first, there's a second local maximum you can move to which infects the person but evades existing antibodoes, and second, every in-between mutation between the current local maximum and the new local maximum must also be a viable mutation. So, influenza's "search space" may contain many local maximums all with robust mutations in between them that allow many pathways and many variations, and that's the reason it's so successful. However, none of that implies that all viruses have this quality. It could be that the "sweet spot" for the current coronavirus to operate on humans is relatively small and that there are other viable genomes it could use which evade the antibodies for the current one, but there are non-valid genomes in between both sets of mutations, meaning one cannot flip to the other through normal-sized mutation and selection.