The “record picker” allows one to search for and select a record by the primary field only, it seems. I would like to see a true search element that can be configured to search multiple fields, or all fields within the specific view.
Yes, that would be a nice feature to have all throughout Airtable — for example, anywhere a linked record field shows up.
You might want to check out On2Air: Amplify, which lets you create custom interfaces that can search across any field:
Also, if you’re using Airtable’s normal built-in views that come with Airtable, you can use Airtable’s search app. (But apps don’t work within interfaces.)