Links still do contribute to local rankings — but they’re far from the be-all end-all they are in traditional organic search. I would encourage your old-school guy to look at this research from Local SEO Guide. The first factor related to links doesn’t show up until #12. Factors 2-11 all relate to reviews. Individual factors shouldn’t be taken as gospel, but directionally they should signal to your colleague that there’s a completely different algorithm at work when it comes to local.
And if he needs to prove it to himself, have him look at the top 10 businesses in any reasonably competitive category of the Google Maps Finder. Say, personal injury lawyer Houston. Run the websites listed for each business through your favorite link tool. I’ll bet your guy a beer there’s no readily-identifiable correlation between domain authority and local rankings.