Useful abstractions. Exploring their usefulness can be fun (the pedantic kind) and even sometimes illuminating.

Obviously right from the start "taxonomy" itself is arbitrary. A more universal tool than naming, establishing branches, starting and end points is the concept of "most recent common ancestor" [0]. From that nameless unclassified vantage point I find it interesting to see (as OP pointed out) how "woodiness" or "treeness" evolutionary speaking is clearly a "strategy" (convergent evolution) and nearly impossible to chase down "phylogenetically" through MRCAs. Intuitively before reading this article I would have guessed this to be more straightforward.

[0]https://en.m.wikipedia.org/wiki/Most_recent_common_ancestor