If the only alternative is between a rewrite, and not-fixing the mess gradually, then I'll take the rewrite anytime and let the cycle continue.

The problem is: is your rewrite really going to be a full-rewrite, or some kind of hybrid monster (at the architectural level, of course, there is no problem in reusing little independent pieces, if any exist)? Because you can easily fall in all the traps of both sides, if the technical side is not mastered well enough by the project management...