2 things
1. LLM can fix this in my experience. an LLM, especially the more modern ones, have way more short term memory than most humans (or at least way more than me). They can dig through this kind of code and figure out all the edge cases, write test, suggest various paths to make things better and then execute on those paths. On request they will happily setup dev systems, staging systems, whatever it takes to make the transition safe. At least that's my experience. They can dig much further than I ever would.
2. Short of and maybe separate to the LLM fix, this pattern of technical debt I think is nearly inevitable, at least with humans. In a perfect world, every human and every reviewer knows exactly what architecture to write and what tests to convey all of the rules and assumptions because no matter what, people are going to leave. I've never seen that code base though. So the rules and assumptions are at best half written down, maybe in some comments or docs, comments or docs that the next person to edit that part of the code base may or may not see. And so it goes.
I work on a code base that runs on Windows, Mac, Linux, Android, iOS. Those OSes change over time, their requirements change, their APIs change, the world changes and new APIs are needed for new things people do, and our original choices for cross platform solutions, no longer fit perfectly. We need to keep moving and shipping and we can't just stop the world and re-architect. Further, like the OP, not everything is written down and communicating across hundreds of programmers is hard. So yes, not every decision is perfect. It never will be. Fortunately we work to fix these things and pay down our technical debt but it can take 5 to 10 years before we can finally remove some old path while we wait on usage numbers for old OSes to go down far enough that we can remove those paths and switch everyone over to the new. And then the cycle repeats.
For 1, effectiveness still depends on the size and complexity of the codebase.
From my experience, as complexity and size grow, each prompt takes longer, does less, and is prone to more mistakes and disruptions to other parts of the codebase.
> an LLM, especially the more modern ones, have way more short term memory than most humans
But it's not unlimited. So they can write code that they "understand" within the limits of their working context. When it grows beyond that, they can't cope with it in its entirety. And a human, at that point, has no prayer.
You are the very first person I encountered who claim LLM can fix technical debt. Usually I seem comments and articles saying LLM can only produce it. And I can agree with those articles and comments