I propose we use the same media controls that we use for the group block's "Background Image" PanelBody, for a "Media" PanelBody within the cover block's Inspector.

It would closely resemble the site logo's "Media" panel for uploading a logo.

This could pave the way for putting the cover block's media in the Styles panel of the Inspector in a future iteration, so you don't have to flip between Settings and Styles to adjust the focal point and change the overlay, etc.

As an aside, the group block's background image popover lacks a bit of finesse. Would be great if they were shared so that fine-tuning one does the other as well.