Yes, dairy provides protein but extracts containing (virtually) no protein also lower uric acid. It’s one of the best-established relationships in the field along with the effect of alcohol.
As to cherries, have you checked Jacob et al. (see the footnotes of the British guidelines)? Possibly it’s a junk paper but some expert thought it was worth referencing. I haven’t read it myself. It’s not the only one claiming an effect on uric acid anyway.
The fat metabolism does play a role in gouty inflammation… again, it’s not a cause of gout but modulates its expression. Keith has a page or two about this on Goutpal.
There is also as you mention the fat-related comordities of gout. Though it obviously isn’t relevant for every individual, certain fats are maligned for a reason regardless of the popularity of some truther barguments.
On top of that, there is weak evidence suggestive of fatty diets contributing to high uric acid. Carbs other than fructose (and of course sugar) seem to give better results in studies. Anectodally, many have come to this forum saying they developped gout after going on a low-carb diet. But this may of course be due to excessive protein intake or simply purines.