Many pathogenic viruses have multiple species as hosts. For example Covid has a long list of species it infects from mink to deer as well a humans. In plants cucumber mosaic virus has over 1200 species it infects and is transmitted by more that 80 aphid species. These are the generally ones we should be the most concerned with emerging from the artic.
There are viruses that are limited to one species. The only reason we were able to eliminate smallpox was because it was human specific. If it had retained it’s ability to transmit via rodents (it evolve from infecting African rodents) it would likely still be around today.
In evolutionary time 20,000 years is not long. With a 25 year average generation time, that’s 800 generations. At an estimated average rate of mutation of 64 per generation that would be 51,200 mutations in that amount of time across 3,117,275,501 base pairs (female).
This might sound like a lot but only 2% of the human genome encodes proteins. So approximately 1,024 mutations to encoding proteins are possible across 62,345,510 base pairs. However we know that many changes to encoding proteins are conserved so the true number is less.
This is really a tiny number of mutations. It may take out a few viruses but most of them will still easily infect humans today. The difference between the mutation rate of viruses and our own generation time is why dynamic immune systems have evolved. This allows an organism to develop immunity to the rapidly mutating pathogens without waiting for changes to the DNA.
As for if this is fear mongering or not: It’s a real probability because of the degree of preservation we have found on carcases in the permafrost. As the permafrost melts these carcases are exposed and transmission is possible.
Honestly the largest worry would for something like influenza. Able to jump multiple species and recombine into novel new types. We also don’t know how long influenza has been around.