[PoC] Change LogExplorerTabs to persist current tab state instead of state sharing

[davismcphee]

Summary

WIP.

Checklist

• Any text added follows EUI's writing guidelines, uses sentence case text and includes i18n support
• Documentation was added for features that require explanation or tutorials
• Unit or functional tests were updated or added to match the most common scenarios
• Flaky Test Runner was used on any tests changed
• Any UI touched in this PR is usable by keyboard only (learn more about keyboard accessibility)
• Any UI touched in this PR does not create any new axe failures (run axe in browser: FF, Chrome)
• If a plugin configuration key changed, check if it needs to be allowlisted in the cloud and added to the docker list
• This renders correctly on smaller devices using a responsive layout. (You can test this in your browser)
• This was checked for cross-browser compatibility

For maintainers

• This was checked for breaking API changes and was labeled appropriately