Your question may be based on a misconception. Mutations occur all the time. In almost all cases, it leads to decrease in reproductive viability. For a virus with one or more established host species, it has already evolved to be quite efficient at reproducing within those species. For it to jump to a new host species and spread in that other species, it must not only be able to infect the new host species, but must also have the opportunity to cross over. If it cannot normally infect the new host species, then it has to acquire some mutations. But mutations occur all the time, whereas the opportunities are not always there. So this is a matter of relative likelihoods. The more contact between wild animals and humans, the greater the likelihood of such inter-species infection happening, because of low prior contact between humans and wild animals.
Since we now have probably 10 million or more humans infected, we have provided a great number of opportunities for the virus to jump to other species, many orders of magnitude more than the original opportunities for the virus to jump to humans. This is why such large-scale events can have knock-on effects of much bigger scale than the events themselves.
While zoo visitors may not go close to large cats, they may sneeze or cough and thereby transmit the coronavirus in mucus droplets that can last very long even if alcohol-based disinfectants are applied. The zoo keepers themselves may be infected and have close contact with the animals. It is difficult to trace the source of infection when we have so many cases. Although the Bronx Zoo was supposed to be closed from 16 Mar onwards, it is not impossible that some animals were already infected prior to that, so any hypothesis is going to be guesswork.
Furthermore, viruses that already regularly infect more species tend to be able to infect more new species than other viruses. So once the coronavirus managed to jump to humans, it is reasonable to expect it to be able to infect other new species more easily than the strain in the original host species. So the fact that tigers are susceptible to the strain that infects humans does not imply anything about whether that strain has acquired some new mutations or not.