Resolve "When changing project visibility setting, change other dropdowns automatically"

What does this MR do?

1. Automatically changes the visibility settings of non-disabled project features to match the changes to the project's own visibility setting.
2. Animates effected dropdown elements to draw attention to the changes made through side effects.

Are there points in the code the reviewer needs to double check?

I've resisted the temptation to throw out and rewrite project_new.js from scratch, but I really think this ought to be remade as a Vue component in the future. As it stands, this file is a complete mess of technical debt that ought to be resolved in a separate issue.

Screenshots (if relevant)

Does this MR meet the acceptance criteria?

• Changelog entry added, if necessary
• Tests
  • [ ] Added for this feature/bug
  • All builds are passing
• Conform by the merge request performance guides
• Conform by the style guides
• Branch has no merge conflicts with master (if it does - rebase it please)
• Squashed related commits together

What are the relevant issue numbers?

Closes #24032 (closed)