The methods say people were allocated to vegan versus "high meat consumption" foods, with participants free to choose their diets. The only restriction was you had to "try" to increase intake if losing weight, and "try" to decrease intake if gaining weight. (i.e. only isocaloric per basal rate, not across the board). Then the linked studies hint heavily that the vegan group tended towards less protein but more fibre, and roughly equal carbs (fat is not mentioned!!). Which at the very least shows the diets were not "equivalent bar for meat", let alone for "moderate" (i.e. potentially healthy) amounts of meat.
So they're not comparing "healthy wholefood vegan vs healthy wholefood meat diets". They're comparing availability and behavioural tendencies towards dietary choices when aiming for a short term vegan diet versus when aiming for a short term diet characterised by excessive amounts of meat.
I honestly don't get why they do studies like this. It's a waste of time for everyone involved. It's almost like they're only after a headline extolling veganism, rather than actually useful comparisons that could lead to real, actionable dietary insights. Really, if anything, a supported conclusion here could even be that meat diets are likely healthier given sufficient fibre supplementation.