Address findings from `dcm check-unused-code`.

[kenzieschmoll]

Command: dcm check-unused-code packages/devtools_app/ --exclude-public-api

• devtools_app/
  • lib/src/extensions/
  • lib/src/framework/
  • lib/src/screens/
  • lib/src/screens/inspector/**_controller.dart: investigate usages of inspector controller internally before removing this
  • lib/src/service/
  • lib/src/shared/
  • lib/src/standalone_ui/
  • all other lib/src/ files
  • integration_test
  • test/
• devtools_app_shared/
• devtools_extensions/
• devtools_shared/
• devtools_testing/