What does first-party change inside the building?
Less than owners expect on the line, and more than they expect in the office. On the pass, a first-party ticket should print, alert and route exactly like a marketplace ticket. If it lands on a fourth tablet with a different sound, service suffers and the staff routes around it. The fix is an order aggregator that puts every channel in one feed.
In the office it changes what you can do on a slow Tuesday. With first-party orders you have a list: who ordered, what they ordered, when they last came in. You can text 40 lapsed regulars about a catering tray. On third-party volume you have a deposit and a rating, and the only way to reach those diners again is to buy them again.
One more practical note. First-party does not mean you drive the food yourself. Delivery is a separate purchase, dispatched per order at a flat cost. Ownership of the diner and ownership of the car are different questions.