Several restaurants I frequent do a robust take-out/delivery business. While it’s good for them, it seems that they have a hard time fulfilling the in-person orders while also trying to prepare the to-go orders in a timely fashion. It’s the modern-day version of retailers trying to answer the phone quickly while still helping the customer in front of them.
Organizations that have multiple inputs may be best served by having distinct processes to handle the different channels. Let a crew handle online orders only. Have someone dedicated to handling chat and online messaging while others address in-person customer needs. Designate a person to answer the phones and another to be a welcoming receptionist. Different inputs require different strategies for successful outputs.

