With pathogens that jump between species, as SARS-CoV-2 has with several populations, including white-tailed deer[1] in the US, you potentially need to isolate all reservoir species from each other as well.
Cross species transmission is rare and while some cases have been reported, they're really the exception rather than the norm.
If there was really that big of a risk where we'd need mammals to isolate as well, then veterinarian services the world over would have collapsed under the weight of house pets contracting the virus.
edit: curious about the downvotes. Perhaps you have proof that has eluded the virologists researching SARS-CoV-2. Please do speak up ;)
Why would they get severe disease when they get infected? Even most humans don't, let alone other species with different receptor setups. The virus is absolutely in non-human animal populations almost everywhere, and is not a candidate for eradication.
With pathogens that jump between species, as SARS-CoV-2 has with several populations, including white-tailed deer[1] in the US, you potentially need to isolate all reservoir species from each other as well.
[1]https://www.nature.com/articles/d41586-021-02110-8