Refactor rerun menu slightly

[emilk]

What

Small refactor. We talked about maybe making a test for this, but in the end we decided to test something else.

Checklist

• I have read and agree to Contributor Guide and the Code of Conduct
• I've included a screenshot or gif (if applicable)
• I have tested the web demo (if applicable):
  • Using examples from latest main build: rerun.io/viewer
  • Using full set of examples from nightly build: rerun.io/viewer
• The PR title and labels are set such as to maximize their usefulness for the next release's CHANGELOG
• If applicable, add a new check to the release checklist!
• If have noted any breaking changes to the log API in CHANGELOG.md and the migration guide

• PR Build Summary
• Recent benchmark results
• Wasm size tracking

To run all checks from main, comment on the PR with @rerun-bot full-check.

To deploy documentation changes immediately after merging this PR, add the deploy docs label.

[Wumpf]

lgtm!