Lately I've been testing a "third way" of thinking about refactor/rewrite decisions, which is to first document the existing requirements.
The trick is in making those docs so small and lean that they fit neatly in comments inlined with key functions, and this I have accomplished by reduction to a list of "must" and "cannot" (abbreviated to + and -) bullet point items. It must function like this, it cannot function like that. Some requirements can be turned into automated tests or static checks, others need an eye to verify. But the cycle I'm seeing often works out to be: write speculative requirements, write speculative code, then reverse-engineer true requirements and maintain code against that.
Doing this creates a base quality bar that the code has to hit in both the rewrite and refactor cases. The trap that rewrites tend to fall into is that the requirements are lost during the rewrite, so the new code becomes just as speculative as the original greenfield code. If you have the requirements as your reference, on the other hand, the area of bad code you can write is limited to the boundaries of those requirements and any missing requirements. This makes new code much less of a gamble.