Then learn their language and phrase your suggestions in ways that expose the issues to them in terms they understand. This requires you to fully understand what is important, and what is not important, to the business.
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.
But is it really my duty to play managers, so they are ok with refactoring and improvements? I imagined somehow, that management and developers have same goals and it’s natural thing having clean codebase and up to date electronics.
Yes, it is your duty to make them understand. Especially if they don't have a background in development themselves.
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.
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.