I'm having the same issue. I've owned Dine Out for years and although it was always a little buggy, it's completely unplayable now. I used to like owning restaurants and taking mine out to eat and I can do neither. I have the problem with the chef not entering, so I thought to try opening the restaurant first, then travel to it. Not only was the chef still standing outside, but the host wasn't seating customers either when I did it that way. When I traveled to the lot first, then opened it, the chef wouldn't make a move to enter until 2-3 hours. Shift clicking the chef didn't do anything, they wouldn't move until at least 2 or 3 hours into their shift.
Then I tried going to the restaurant as a customer and none of the staff worked. They wouldn't go to their stations no matter what I did. It never used to be this bad, I used to have fun with different kinds of restaurants and learned how to furnish them in a way to make them as functional as possible. But not anymore. It doesn't matter what I do, if it's not one staff member, it's another, or all of them.
I installed Carl's mod yesterday and there is a noticeable improvement when my sims are customers. Then ordered, got their food quickly, I thought the mod would be a good solution until the day (if it ever does) comes when the pack is fixed or gets a refresh.
But it is not the case as a restaurant owner. Shift clicking the chef does make them work, but then they go in an invisible cooking animation loop forever and the customers never get their food. No food ever appears, they fiddle with the gas, and stir invisible pots for hours and hours. Then I get angry customers and no money because they never get their food. I tried with different restaurants, different staff, nothing works when I try running a restaurant. It's unplayable and incredibly frustrating.
However, Carl's mod does help a lot with dining as a customer but it seems with every new pack and every new patch, Dine Out just gets more bug riddled than ever. Nobody should have to go through this much trouble to get something to work that they paid money for. A newer player might think that something is wrong on their end rather than it being a game bug. And of course console players don't have the option of using mods, it's just not right. I'm very frustrated, I've been trying for days to figure out a way to run a restaurant again and it's just not happening. It's not worth the aggravation anymore and it's a feature I used to enjoy.