And unfortunately nothing will change because what you want to do is better in every single way, but their way of doing thing is better in the only one that actually counts: It's faster and cheaper.
My assumption is that I will never clean up this entire codebase. Rather, I’m looking for the best way to triage the situation: what are the things I can/should try to fix first?
Make a list and organize all the things you'd want to change. Untangling balls of mud often take on the order of months to start seeing any kind of progress. You have to make sure you keep things organized over a long time otherwise you can't tell whether or not you are making progress.
If a module breaks often when seemingly unrelated things change or breaks every time someone touches it it's easy to say the module is badly written. It's often hard to say why it's bad and harder to say how to fix it. However at least you will have data and be able to defend your position that messy code in a particular module actually slowed down the addition of a new business features.
If you are using a bug tracking tool (Jira, Bugzilla etc) make a bunch of private ones for yourself. If management doesn't like all your private tasks showing up on reports then try to get a section for yourself or see if you can deploy your own "private" bug tracking instance.
Any time you encounter a problem make a bug for it even if you have no solution. If upgrading a library version a minor version causes a break make a bug for it etc. You'll start to see patterns and then you can make small but really important surgery on the code. You can have a plan for how to fix a particular module when the time arises.
For example we had a core module with some 190 downstream internal clients across some 5 different dev teams. A recent Oracle version upgrade caused problems because we would have to to coordinate everyone to update at the same time. I already had a long time standing issue that the module is poorly organized and already had a list of all the things that were wrong with it. When this issue came up I immediately stepped in and how to completely remove Oracle from the core module so that the downstream clients can each upgrade on their own time. Because I had thought about this for time I was ready to defend the proposal by showing it would take less time to do this. I was able to do a refactoring I've been wanting to do. Over time people will start to listen to you more if you are actually delivering shit on time. Otherwise, even if your complaints are correct, you will be labelled as just a whiner.
Write down everything you know about the codebase. Whenever you discover a surprising dependency or behavior make a note of it. This is something you can do in passing as you push features.
At some point you'll start recognizing structures in the code some of which will be small enough to surgically improve.
