The quick inserter currently surfaces patterns, both by default and when you search:

But it's not always clear whether the options (like in the screenshot above) are template parts or patterns, which can lead to folks mistakenly inserting a pattern where a template part would be preferable.

This issue can be partially fixed by #41397, but it might be best to establish some visual distinction to cover cases where patterns and template parts share a name. There are a couple ways to do this, one solution would be to simply list them separately:

This would be similar to what you find in the current implementation of the template part replacement modal:

Another option would be to list them together but use the template part icon to distinguish between patterns and things like headers / footers. Rough example: