That element of it — when the restaurant is doing well, the windfall is shared with the waitstaff — could be preserved by simply giving the staff a percentage of the price of each meal they work on. Structure it as a bonus, the way salaried professionals can receive a bonus when the company is doing well.
It may be worth noting that worker-owned restaurants, like Cheese Board Pizza here in Berkeley, typically do not solicit tips. (Well, except for the live musicians, who are not worker-owners.) If tipping was really all that great for the workers, then places where the workers literally control company policy would encourage it.