If it's in the codebase for a single project, then sure, you can do some refactoring while you work, add more tests, etc. It's not too big of a time sink if you do a little at a time, and you can do it all by yourself.
But if the tech debt is in the architecture, or the API spec, or the authentication layer, or in some other area that impacts how services owned by multiple teams communicate? Now you need coordination with other teams to fix the debt. Everyone else has their own jobs to do, and if leadership doesn't make fixing tech debt a priority, other work will always take precedence.
I agree coordination is needed for cross-team work, but why must that involve people on the product side of the house? In the same way a team can internally prioritize particular technical work, the technical people can coordinate between teams.
I agree that if a mess is particular large, or cleaning it up will have product/schedule impact, sure, the you have to talk with the product people. But the converse it true: if you can find ways to do the necessary technical stuff while still producing sufficient business value, then the product people will never notice.
If it's in the codebase for a single project, then sure, you can do some refactoring while you work, add more tests, etc. It's not too big of a time sink if you do a little at a time, and you can do it all by yourself.
But if the tech debt is in the architecture, or the API spec, or the authentication layer, or in some other area that impacts how services owned by multiple teams communicate? Now you need coordination with other teams to fix the debt. Everyone else has their own jobs to do, and if leadership doesn't make fixing tech debt a priority, other work will always take precedence.