From the conclusion of the paper you linked:

This review has found that there is no convincing evidence of major impacts of vegan diets on dog or cat health. There is, however, a limited number of studies investigating this question and those studies available often use small sample sizes or short feeding durations. There was also evidence of benefits for animals arising as a result of feeding them vegan diets. Much of these data were acquired from guardians via survey-type studies, but these can be subject to selection biases, as well as subjectivity around the outcomes. However, these beneficial findings were relatively consistent across several studies and should, therefore, not be disregarded.

There is an urgent need for large-scale population-based studies to further investigate this question, with a particular focus on assessing the dietary aspects cited to be of particular concern, e.g., taurine and folate. For guardians wishing to feed their pets vegan diets at the current time, based on the available evidence it is recommended that commercially produced vegan diets are used since these are less likely to lead to nutrient imbalances.

While it does support the viability of specially formulated vegan dog and cat diets based on the current research it is important not to gloss over the fact that they also stress that the current research is lacking and largely based on self-report surveys. Personally I'm not terribly swayed by this paper one way or another and wouldn't take it as being definitive. Of course I recognize that more precise research has difficulties due to the ethics involved, but I'm also confident that we can do better.

I agree with what you say about the obsession with natural diet being weird by the way, but I think there is a reasonable disconnect in the leap from natural meat -> meat based pet food ------> no meat. For example, even if I don't eat the same food an early homo sapien would eat I still eat the same kind of food rather than an all mineral diet or something. That's not to say that I wouldn't if such a thing were viable of course, just that I'd want to be very sure first.