Yes, I've seen it cleaned up. I've even seen (and been on, and led) teams who track their technical debt, measured in terms of estimated person-days to clean it up, and deliberately schedule periods of time focused on debt paydown, both short sprints and longer-term efforts.
I've even seen organizations mismanaged like yours which have turned around and began deliberately managing their technical debt. It starts with the engineers educating management about technical debt. Unless the managers are fools (and most of them actually aren't, they just seem that way because engineers and managers don't communicate well), they quickly grasp the idea and its consequences. Even then, you can still have managers that care only about the short term, because they expect to be promoted and out of the mess before the note comes due, and there's really not much you can do about them except to quietly try to manage it yourself. But given a reasonably-intelligent manager who is interested in next year and the year after, teaching them (gently, non-confrontationally and above all non-condescendingly) about the concept of technical debt will help them to understand and work with you.
One key to making it work is that you must be able to quantify technical debt. That means being able to make reasonably-accurate estimates of its impact and cost to fix. That, in turn, means that you have to be capable of making reasonably-accurate estimates of how long it takes your team to accomplish a given piece of work (feature or debt paydown). And you have to establish a track record with the manager so that he or she trusts you and your estimates.
I'm not saying it's easy, but it absolutely can be done... and it'll make your time at work much less stressful. It may or may not result in any decisions being changed, but when you're sure that management understands your concerns, and considers them to be perfectly valid and important, it makes it much easier to accept when they decide to take the shortcut anyway. And if the team is visibly and conscientiously tracking technical debt (e.g. as a regular part of weekly status reports), most managers will begin seeing it as a variable to manage, including by finding opportunities to pay it down, and by understanding when it impedes new work.