So, the question is, how do you deliver better code faster?
I think there question needs to be: how do we convince people that it's "worth it" to pay for better code?
I get it, we all want to do our jobs well and excel in the craft. The problem i see here is that the stated need for cleanup, refactoring and documentation is immediately followed with a 'but'. But there is no but, imo.
Code that "works" isn't good enough, and we need a full stop right there and heads around the table nodding up and down in unison. But they'll take the inferior POC build every time over investing in a "good" solution that incorporates lessons learned. Next they complain that it can't be modified easily ("but, it's just a simple change!"), and bemoan the state of software development.