ListItem 2.0 (part 0): re_ui_example refactor (#6148)

### What

The first stage of the `ListItem` 2.0 effort will be to deploy on
`re_ui_example`, which is currently a mess. This pure refactor PR splits
it into multiple file as a preparatory step.

### Checklist
* [x] I have read and agree to [Contributor
Guide](https://github.com/rerun-io/rerun/blob/main/CONTRIBUTING.md) and
the [Code of
Conduct](https://github.com/rerun-io/rerun/blob/main/CODE_OF_CONDUCT.md)
* [x] I've included a screenshot or gif (if applicable)
* [x] I have tested the web demo (if applicable):
* Using examples from latest `main` build:
[rerun.io/viewer](https://rerun.io/viewer/pr/6148?manifest_url=https://app.rerun.io/version/main/examples_manifest.json)
* Using full set of examples from `nightly` build:
[rerun.io/viewer](https://rerun.io/viewer/pr/6148?manifest_url=https://app.rerun.io/version/nightly/examples_manifest.json)
* [x] The PR title and labels are set such as to maximize their
usefulness for the next release's CHANGELOG
* [x] If applicable, add a new check to the [release
checklist](https://github.com/rerun-io/rerun/blob/main/tests/python/release_checklist)!

- [PR Build Summary](https://build.rerun.io/pr/6148)
- [Recent benchmark results](https://build.rerun.io/graphs/crates.html)
- [Wasm size tracking](https://build.rerun.io/graphs/sizes.html)

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