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.