Sorry, my engineering my mind craves a solution. Wouldn't forcing people into an asynchronous online/mobile queue be the best solution? You can favour verified locals if you want there, control exactly how many orders you take in, and nip any potential disappointments (leading to bad Yelp reviews) right in the bud.
I've trekked to enough "Best <X> in <Y>" places and been the "outsider" grumbling waiting outside in a giant lineup (or being at the front of the line at 9am), to know that it is a cherished experience while travelling, giving you a piece of local flare, and like many problems a focussed effort could help it scale properly.
I feel like the problem there is that it would require expending a lot of effort to solve what will probably be a temporary problem. That kind of system isn't going to be cheap, and the high traffic will only last as long as the list in question is popular. As soon as another 'best burger place' comes along, all that traffic is probably going to die down, and now you're a small diner left with this complex expensive system.
It reminds me a lot of the dilemma facing graphics card makers recently, wherein the cards keep getting bought up by cryptocurrency miners. If they raise prices, they're pricing out of the regular customers who will stick around after the cryptocurrency mining dies down, and if they raise their production capacity too much then they will be left with massive overcapacity with things die down.
I'm thinking this could be a solution that various businesses could adopt temporarily. And not a turnkey solution for each business.
These stories also keep coming out of various businesses in Europe with the influx of Chinese and Indian tourists, eager to pay them but the businesses have no infrastructure to "shape" the inflow of customers.
If this is indeed a big problem, this is also a big opportunity.
You're trying to engineer a solution to the problem of "how do you make everyone happy" (albeit in a specific scenario). If you change things, someone will be upset by this. At best you can hope to optimize this to make the MOST people happy and the LEAST people unhappy, but you can't get a perfect score here.
I've trekked to enough "Best <X> in <Y>" places and been the "outsider" grumbling waiting outside in a giant lineup (or being at the front of the line at 9am), to know that it is a cherished experience while travelling, giving you a piece of local flare, and like many problems a focussed effort could help it scale properly.