How do backlinks to a GMB profile connect to the map pack?
Google says prominence is one of three main local ranking inputs, next to relevance and distance, and that it includes information from across the web such as links, articles and directories. So links belong in the picture. How much they matter in a given city and category is not published.
A link on its own does not turn a thin profile into a strong one. It adds to a base that has to be in place: the right categories, accurate details, real reviews and a site that matches what people search for. See our guide to map pack ranking factors for the wider picture.

