Also keep in mind that often the existing application is the only real artifact you have capturing all of the complexity and corner cases your rewrite will have to deal with.
Sometimes you manage to catch some of these early on, but often they aren't caught until later in the process -- at which point the cost for the change (and the impact on budget/timeline, and potentially even the very architecture you set out to fix!) is drastically higher.
I've seen it happen a number of times where the 'spec' seemed simple enough at commencement & throughout most of the project; then you'd have each department test & the reports would start coming in... and the result would inevitably be a multiplier on the project scope.
Not from missing features mind you, but via the omission of 'complexity handling' / business logic which was documented -- but the code itself was the documentation, which isn't overly unusual (and IMO isn't even a particularly bad thing in many cases, though not all).
Code-as-documentation is best-in-class at staying up to date, when it comes to documenting what a system does and how. Often the relevant pieces are very discoverable as you're going to make a change. For other goals, it's much less discoverable. Clarity depends on both how the software is written and who the audience is.
Tests-as-documentation are arguably a special case of code-as-documentation. Outside of the case of well maintained tests-as-documentation, code-as-documentation often has a hard time expressing "why" and distinguishing between what has to be a certain way and what just happened to be that way.
It also has some trouble expressing aspiration - "we've decided it should be this way, but it's not yet".