This does seem to be the case. Pathogenic (or even nonpathogenic 'silent riders' with minimal effects) respiratory viruses exist in a wide variety of animal species, and it seems the main barrier to cross-species transmission is that the natural host's cell-surface receptor proteins are different enough from human cell-surface receptor proteins that transmission is blocked.
With all the modern biotech tools, it's now possible to swap out the cell-surface binding domain from any virus with one that matches a human cell-surface receptor protein, which is how the viral particle gains access to the cell's interior. Cellular interiors seem more highly conserved across species, i.e. it appears that once you breach the outer defenses, the rest of the viral package can replicate within the cell without being constrained so much by inter-species differences. The ribosomal machinery will build just about anything in other words.
Hence the 'gain-of-function' game, which proponents justify as 'finding the potentially dangerous mutations before they arise naturally' is just creating novel pathogenic viruses with a high chance of escape from the labs where they're being created. This kind of research was temporarily banned in 2014 in the USA, but then that ban was lifted in 2017, all with little public discussion.
A permanent international ban on this kind of research, including an inspection and monitoring regime, is likely the only long-term solution that will work.