`ErrorBadgeManager` rewrite to separate inspector specific logic

[elliette]

ErrorBadgeManager has a lot of conditional logic for inspector errors (e.g., appendError only works for inspector errors because we are only saving inspector errors in the `activeErrors map: https://github.com/flutter/devtools/blob/36ac0a283cf1c1265f786f5fde513853126760ca/packages/devtools_app/lib/src/shared/managers/error_badge_manager.dart#L32-L37)

This file and logic could use an overhaul to move the Inspector specific logic to the Inspector controller.