Fix deletion of `output.overrides`

[tbrockman]

Details

Fixes an issue with overrides key not being deleted when the browser target doesn't match 😳

Adds testing in examples/with-browser-specific-overrides to verify manifest.json is produced as expected, which should probably be merged before this PR (and then this one can be updated with the new ref).

Adds test runs to CI (I think).

Code of Conduct

• [x] I agree to follow this project's Code of Conduct
• [x] I agree to license this contribution under the MIT LICENSE
• [x] I checked the current PR for duplication.