"I'm curious now, how should food/drinks get delivered if not via tray? I always thought use of the tray was more professional and "cleaner", as well as no chance of anyone's fingers inadvertently going in my sauce."

The waiters can carry it to you. Many people in the industry feel the same and insist on staff using a tray with the end result being the OP. The waiter could have brought the drinks by hand. The waiter has to pick up your food anyways, so if you're worried about fingers in your food then why eat out. Any professional will have 0 chance of putting a finger in your food. Plates have a brim for a reason.