If we had reasonable social safety nets, I would say quickly that he is irresponsible staying after finding out about the code base. But the problem is that employers upsell the stuff you're going to be working on. They don't tell you upfront that it's shit and you're the one to rewrite it. If you ask them about the quality of their code, they'd likely respond with "it has some painful points here and there" or some other non-answer.
If they hired OP to rewrite it, then yes, they're not fulfilling their responsibilities. But it sounds like a place that just shovels JIRA issues at you and tells you to go fix them.
An employer-employee relationship does not necessarily imply "I must do everything in my power to make sure this product is the best that it can be within my 40hrs/week". If you are leadership or a product owner, you have that burden, but not necessarily a LOB developer position. If they gave him that responsibility, they should immediately ask for a significant raise, because they've changed virtually the entire job which they were hired for.
