The science in the paper is fairly interesting. It observes that:
1) Certain types of mutations are favored by different host biologies. In other words, in humans, you would tend to see one type of coding substitution, in mice, pigs, bats, etc. you'd see another type of coding substitution. The paper gives specific mechanisms that underly this biases; I'm not an expert in these, so I can't say for sure if they are correct, but it the mechanisms as proposed make sense on a first read-through.
2) The early omicron variants have a set of mutations that would be consistent with evolution in a non-human host, overlapping most strongly with mice biology.
3) Perhaps even more significantly, _later_ omicron variants that have spread through human populations have picked up mutations that are once again more consistent with human biology. So the paper manages to show both positive and negative data toward their hypothesis, which makes it stronger.
For the TL;DR crowd, figure 3 summarizes this pretty well.
To analogize this in computer terms, it's as if someone was tracing the evolution of memes by observing the types of compression artifacts introduced in various re-compressions of the meme. For example, JPEG might introduce artifacts related to DCT math, whereas JPEG-2000 would introduce artifacts related to wavelet math. The analysis shows that the artifacts of this new Omicron meme has compression artifacts consistent with, say, Wavelet compression up until it jumped to being shared by JPEG-users, at which point newer modifications to the Omicron 'meme' show JPEG-consistent artifacts.
That's basically the scientific core of the paper. So, to contest the validity of the paper, I think it would be helpful to find citations that dispute the observation that mutations accumulate with a certain bias based on the host biology, rather than whattaboutism pointing to other hypotheses and leaving it at "there are other alternative explanations" therefore this paper is not interesting.
So, while I don't dispute your statement that there are other alternative explanations, what I would really like to see are comments addressing the core scientific thesis of the paper, which is that certain mutations appear more frequently in other host biologies; and that the first omicron cases showed a preponderance of these differently-biased mutations; and that since it has been circulating in humans, the later mutations are once again consistent with a human host.
At the very least, I thought it was a very interesting analysis, as I had never seen the molecular bias of mutations used as a proxy for host biology before. If this is true, then viruses that can spread through alternate host biologies would gain a diversity of mutations that would be less available co-evolving with a single host, which could lead to evolutionary advantages over viruses that are unable to hop between hosts.