One profile per location, managed as a set
Every location needs its own Google Business Profile with its own address, phone, hours and photos. Managed individually by each general manager, they drift — different categories, different attributes, hours nobody updated.
Manage them centrally, but keep the content local. Consistency of the structural fields, variety in the photos and posts.
The duplicate location page trap
A location page produced by swapping the town name into a template is the most common failure in multi-location restaurant sites. Google treats near-identical pages as one page and picks a winner, which is exactly the flagship problem you were trying to solve.
Each location page needs things that are only true there: the neighborhood, parking, which room can be booked, the manager's name, what's different about that kitchen. If you can't write two paragraphs true of that location and false of the next, the page won't hold.
Concepts are not locations
A group running several brands should not treat them as one entity with branches. Different concepts serve different customers and compete with different restaurants — they need separate positioning, and often separate sites.
Merging them under one brand umbrella is a common and expensive simplification.
Brand search versus discovery search
Someone searching your name already knows you and just needs the right location, hours and an ordering link. Someone searching a cuisine and a neighborhood has never heard of you.
Those need different pages. Most group sites are built entirely for the first person and wonder why they get no new customers from search.
Fix the weakest location first
The largest available gain in a group is almost never the flagship. It's the location performing below its own market — where the demand exists and something specific is blocking it.
Rank your locations against their local competition rather than against each other. The order usually surprises people.