FWIW, the lead author of this paper is an advocate for plant-based animal diets. Doesn't necessarily make the study less valid. The data is what it is. But knowing the author is already of a particular mindset going into the study is worth understanding. Also worth noting, the paper is based on people filling out a questionnaire and is not a clinical study. Again, doesn't mean the study is not valid, but I think it is good to understand.
My take away from reading it is that it seems like dogs can do well on an appropriate plant-based diet. But I think it is overstating the results to say that plant-based diets are better. At best I think you can say they are no worse than other diets. But even then, the sample size of respondents whose dogs were on a plant-based diet was much smaller than the other diets. They controlled for that statistically, but even so the study only included 336 plant-based dogs, which does not seem like a particularly large sample size.