At what point do you finally demand a total rewrite?Solved

Oh, I feel your pain, @amelia. But whatever you do, do not advocate for a total rewrite. The Big bang rewrite is a trap. Management will approve it today, but six months from now they will get impatient because zero new visible features have been shipped. Then they will just pull the plug on your team. 

Instead of starting over, you need to isolate the worst parts. Stop asking management for refactoring time because executives just hear that as developers wasting time. Tell them straight up that delivering the Q3 integration safely without crashing our core platform requires rebuilding the API gateway. Look into the Strangler fig pattern. Do not rewrite the whole app. Just build the new integration properly on the side and slowly reroute traffic to the clean code. You can strangle the old messy code one module at a time until the legacy stuff can be safely deleted. 

@noah-blake is exactly right about the rewrite trap, but I will take it a step further. Technical debt is not always a bad thing. 

Think about why that code exists. If that messy duct taped startup code allowed them to launch fast and get acquired by your parent company, then it served its purpose. It bought them speed. 

The problem is you are trying to clean up the entire codebase right now. Do not do that. Only fix the code that actively causes you pain. Is there a messy legacy module that nobody ever touches and it runs fine? Leave it alone. Let it be ugly. Is there a module that your team has to manually restart and patch at 2:00 AM every Sunday? That is your real problem. Refactor that piece immediately. 

Stop trying to make the code pretty. Just make it survivable.