This may not work for you if you are definitely going for cubed with all sides browned, but if you just need chunks of potatoes that can stay on the grill without getting trashed, go Greek! If you want, peel the potatoes first, or after while hot with those thick rubber gloves BBQ guys use (faster), but boil them until cooked thoroughly. I'd say remove the skins if doing this method, they will not hold up on the grill like the potato will. If anything trade the parboiling, drying and oven labor for just the peeling labor. Cool them outside of the cooler whole so they do not get water logged and can dry, but not too dry. Don't cut hot, they turn to mush on the bottom, crusty on top. Cut on a box grater slicer surface, this goes really fast. Season. They hold up on a grill for a long time because they don't dry out like they would from your baking process, are moist when they hit the grill, and they brown nicely on the bottom. This is also the simplest and cheapest labor for very high volume homefries.
Typically the Greeks will load the grill up before the rush, and then turn it all over, make a pile that you take from at the bottom, and then lay out a whole new layer on the rest of the back of the grill, or use a designated side of the grill and take from what is towards the center/work area, and lay new towards the wall of the grill. You can lay down a really thick layer and put a really big press on them, like I've seen used for 20 steaks at a time to speed up browning during service to renew the take pile, and they do get hot because since they are precooked and have their moisture, the steam wicks up, although it is slower than a thinner layer. They are already cooked, get hot fast, brown nicely, for the same reasons really good fish and chips require the parboiled chips, and it is really really hard to overcook them. You can keep turning them over if you need to which only makes them better, but they are good with just browning deep on a side and having been turned. It sounds like it could work nicely for your workflow since they need to sit around and these get better as they sit, and you have the pace to keep them renewed, while having a ton to grab from without the hassle of dealing with an oven. Typically by the time they are served it is like a really coarse hashbrown that is browned nicely on the bottom, sometimes a little within, and on the top from turning over and mixing as you go, but stays fluffy and soft inside. You can keep turning them over and over if you have too, or pile them onto newer you just turned, and they won't overcook during service. I don't know how this would work with your ultimate potato dish. I guess pull a pile of spuds and grill the veggies with 'em like a fritatta and take it from there.
This is how tons of East Coast diners, and most Greek family restaurants do 'em. This is the standard homefry of Western NY from Syracuse to Buffalo, and they do it like this is in (some or all?) Philly joints too. I'm sure it is like this in many other places like NYC and Jersey, but I won't officially speak for places I haven't been to. Greeks are the cheapest owners I have ever worked for, and they usually base all of their methods on never having to throw something out, because usually they won't even if they should. I've never seen home fries done like this need to be thrown out. You get a feel for the rhythm you need to keep them consistent. I've only worked for one restaurant anywhere around here that did not do it this way, and it was cumbersome and slow and really did not work for high volume.
There's a guy with a little place near me, seats a little over 50, his menu says shallot infused home fries. I don't know what that means exactly. Just a thought. Sounds good on a menu. I'm sure he only uses like one or two shallots and just calls it that.
But that gets me thinking you could roast garlic, saute onion, add to some (veggie or grapeseed) oil, throw into a blender with some other seasonings, and premix all ur spuds with this for a familiar but different (better?)taste. Maybe powdered dried herbs like thyme, especially thyme, and terragon would be great, but I have no idea how these ingredients or their flavors would hold up on the grill. They may get bitter, but I suspect thyme holds up. Thyme, onion and coriander is a good flavor mix, but I don't know if it would look good. There is a Tex-mex place near me that uses cumin, very effectively, but that may not work for your place except possibly in a really low dose. The Greeks usually take each tub of home fries just cut and throw in their seasonings and oil and mix like crazy, but usually forgo the salt and pepper until they're on the line. Black pepper usually loses the volatile compounds that taste good through the cooking process and always tastes the best the less it has been cooked, although in many dishes it does need time to mix in its flavor. Hence the old season as you go mentality. Presalting the potatoes just draws water out and makes the cooking process weird on the grill. Many use the bacon grease but not all do. In my experience whole or coarse dried herbs do not work well for this approach. Overall it is low effort, has a lot of margin for error, they can sit a long time and but get hot really quick, and you can always be ahead on spuds ready for the line without hassling with ovens. I have noticed a huge difference between using these potatoes and leftover bakers. These will hold up better.
Also, I would not bother with reds if you go this way, which may also save you cash. Russets get so much starch washed away that they act pretty much the same, and the peeling is easier. Flavor is another story though, but honestly once any spud sits on the grill long enough, there won't be much difference. I can tell you russets taste better after a long time on the grill than reds, they get too earthy in my experience.
Just some thoughts. I tried the white pepper suggestion earlier tonight, it is a solid one. One last note though, this works better for a standard flat top. A snap-action grill WILL overdo these if you don't watch your temp and pace. Use a side, not the back and keep temp lower there in that case. These tend to never overcook on a manual because the grill slows down with a huge layer of fresh spuds put on it and takes awhile to get back up to and saty at temp for long enough to damage them during rush hour. Peace.