Could you, please, elaborate more about that ugly spaghetti monolith? I have similar situation at work. Maintainability is hard to measure, so my every try to measure and improve stuff ends with words “not a business case”.
Oh, and you should get good at incremental refactorings, and at downscoping your improvements so that they become feasible and easy to sell. Or even not require any selling as they can be done as part of normal work. It is not the big rewrites that get done, it is the small improvements that take only a little time, so learn how to compound them.
By the way, getting good at the above points will over time also change your own priorities.
Also I would not frame it as playing them. That is the wrong mindset. You should be among for a high level of trust from their side and then playing them can backfire.
You might also want to read Refactoring to Patterns, but the legacy book is more important to start with.