My impression: it's still easier than a rewrite. People always underestimate a rewrite, because they only consider the complexity they can think of, which is generally only the tip of the iceberg. I've been on projects where even though people nervously joked about the dangers of rewrites they still underestimated the costs. Bonus points if afterwards people scratch their head thinking "where the heck did all that time go"?

I think rewriting to a new architecture piecemeal is probably easier than in a big bang - assuming the thing is at all complex and you can't actually understand all of it at once. The difference is more one of perception. It's easier to see the costs of the Frankenstein architecture than the rewrite, so we overestimate the costs of the former and underestimate the latter.

The real case for a rewrite is if you honestly think all that old stuff really has mostly just sentimental value; i.e. that it's OK to break all kinds of workflows because there are enough alternatives. If you can sell actual users on relearning all their habits, you can get away with a lot.