Lean 4 is not a theoretically-interesting proof assistant. If you're interested in such things, look into Rocq (which uses CoIC, like Lean, but is more rigorous about it), the HOL logic, Isabelle/HOL's automation suite (though Isabelle proper is fairly mediocre, apart from being the thing everyone's standardised around), Lean-auto (https://arxiv.org/abs/2505.14929), and whatever SAT solvers are state-of-the-art this week. Like the tools for symbolic integration and frequentist statistics, there isn't any magic: the power comes from handling enough uninteresting special-cases that we get broad coverage. (Personally, I think there's still a lot of power being left on the table by using overly-general algorithms: sledgehammer is used to crack a lot of nuts, even when that takes quadratic time or longer.)

While CoIC has recursion "baked in", HOL does not. It turns out that we can treat structural recursion as a derived property, even over coinductively-defined types. We don't even need a notion of ordinals for this! (See https://www.tcs.ifi.lmu.de/staff/jasmin-blanchette/card.pdf and https://matryoshka-project.github.io/pubs/bindings.pdf.)

Lean 2 used HoTT, which was theoretically interesting, but not enough was known about HoTT at the time (in particular, whether it was a constructive logic – I think we have all the pieces for an explicit construction via cubical type theory now, but I don't know that anyone's put the pieces together), so that direction has been mostly abandoned. I think there's useful work to be done in that direction, but with the current state of HoTT pedagogy, I doubt I'd ever be able to keep on top of it enough to contribute; and with Lean 4 taking so much of the funding, I don't think we'll see much work in this direction until HoTT is easier to learn.

I still think you're overgeneralising. What actual thing does your poetic tree / thought / river analogy correspond to?