Fix icon picker file state after selecting None

[FuZoe]

Summary of the Pull Request

Fixes the icon picker state transition when switching the profile icon type from File to None and back to File.

References and Relevant Issues

Closes #20307

Detailed Description of the Pull Request / Additional comments

The icon picker now only caches real image icon paths when leaving the File icon type. The none sentinel, built-in glyphs, and emoji values are no longer treated as restorable file paths, so selecting File after None keeps the File controls visible instead of being inferred back to None.

The control also raises the derived icon-type visibility property changes directly whenever the current icon type changes.

Validation Steps Performed

• Added LocalTests coverage for the IconPicker File -> None -> File state transition.
• Added LocalTests coverage for restoring the last real image path after switching away from File.
• git diff upstream/main...HEAD --check

PR Checklist

• Closes Settings UI: "Browse" button disappears/fails to appear when switching Icon dropdown from "File" -> "None" -> "File" #20307
• Tests added/passed
• Documentation updated
  • If checked, please file a pull request on our docs repo and link it here: #xxx
• Schema updated (if necessary)