We need use cases for that in order to start extending the algorithm to include it. My first thoughts would be that pickers and permissions are opposed: if you have a picker, you probably have no associated permission.

If I had to make a pure theoretical exercise and try to have pickers fit in the current model, I think it should be something like granted the reason being that prompt will send the wrong signal to the web page, making it think the permission can be granted by the user.

This said, the only picker commonly used on the web is for <input type=file> and having a permission associated with that doesn't sound right.