It does also backfill the application insights instance with a tag that has the name of your AppService instance, however, other than it allowing the UI to know that you've linked it to an AI instance, I've not found that it actually does anything.

It does also backfill

I'm sorry, what is the "it" here? What does the backfilling and when?

The UI function backfills some tags into the Application Insight instances so the that App Service UI shows that it's connected and has a link, but it appears to be more of a UI hint than a technical restriction.