Convergent evolution doesn't make evolution non-tree-like, that's just a tree with similar-looking branches or leaves at different locations. Pretty standard really. Convergent evolution makes taxonomy less tree-like.
Carcinization is a good example: it makes lots of things crab-like, so from a surface taxonomic point of view a flattop crab, a coconut crab, and marbled crab are all pretty crabby. But they're completely different evolutionary lineages, and only one of them is a "true crab".
> Convergent evolution makes taxonomy less tree-like.
Yes, exactly. "All taxonomies are don't broken" is not at all "incompatible with the theory of evolution," is the point. The taxonomic layer is pasted on top.
Carcinization is a good example: it makes lots of things crab-like, so from a surface taxonomic point of view a flattop crab, a coconut crab, and marbled crab are all pretty crabby. But they're completely different evolutionary lineages, and only one of them is a "true crab".
Where evolution gets less tree-like is MGEs (https://en.wikipedia.org/wiki/Mobile_genetic_elements) as some of them can move between branches.