Would it be advisable to partially pre-grill hamburgers and freeze so that they can be pan fried when a customer orders?

I understand the challenges of running a small restaurant, I've experienced the extremes of the problems. During tourist season I always hired help, but during winter it was just me - Bartender, waitress, cook, dishwasher, bookkeeper, janitor - you name it. Occasionally buses would pull up and unload 30 people on me at once, of course I was never prepped for that. I had some tricks up my sleeve just for that occurrence.

As far as par-cooking your burgers in advance, don't do it. You will lose juiciness, I know this from experience. Besides, if you fully cook par-cooked burgers from frozen, you're really not saving any time after they're ordered. Grill marks don't matter, you don't see them anyway. You're much better off being sure that the patties are of excellent quality to begin with.

The scope of your question is such that all I can really say is, "Don't do it." I do have about about a gazillion things I could share with you that might help speed up your service, minimize waste and put out a great product. Meet me in chat, I'm here quite a bit. You could also write questions that are a bit more broad. I'd love to help in a meaningful way.

This is where low temperature cooking (most people call is sous vide, but it is usually a misnomer) can be your friend. You could cook the burgers until done. Chill. Then flash off on grill or even in fryer before service.

We do large BBQ events for construction companies and need 350 burgers done and served in 15 min. We cook to an internal temperature of 155 and then store in beef stock in the fridge. We then toss on the bbq to bring up to temperature.

If the main issue is speed, consider cooking two thinner patties vs. one thick burger. It's easier to overshoot medium rare, but for the more well done burger it'll dramatically speed up your cooking time.

You can also heat a second smaller pan or a cast-iron press, and drop that on top to help speed up the cooking.

(personally, for home, I'll make patties and freeze them raw, but cook them on a sandwich press ... so it cooks from both sides at once. For a restaurant, you might not have the space to deal with extra appliances, and you'd still have to wait for it to pre-head)

We par cook lots of things, often up to about 90% done: chicken drumsticks, sausages, meatballs, and meat patties (hamburgers?)

They are not exactly the same as freshly cooked, but in their normal serving style (covered in sauce, and between other things) they are not noticeable. The freezing/reheating process tends to render out more fat is the main observation

Test some out, and see if you would stand by them in your establishment

I think a better sauce and a better bun are worth more investment to get customer satisfaction

Ohh, and a really tasty slice of beetroot too :-)