If the average queen mates with an average 15 drones you'll need 750 drones per week to mate with your new queens.
In the spring I feel like I have at least that in each hive. Depending on how many hives you have in your area, I doubt you'd have a problem on that front. I think I recall experts saying that 10% of a spring hive can be drones. Some reports say around 2,000. Either way, that's a LOT of drones! If you're worried about it, throw a drone frame in your best hives to help spread the genetics.
But to the point crofter brought up, grafting and pulling 50 cells in a single day could be a lot to handle depending on what you're doing with them. I tend to do rounds of 30 or so, using double screened boards. This allows me to combine the colony when I'm not grafting and then I can split it back out at my convenience. I just shake all the bees into the top where I put my grafts. All the field bees head back to the bottom. Gives me a ton of nurse bees who think they're queenless while maintaining warmth and convenience. You can also exchange the double screened board for an excluder and you have an instant finisher.
What's your planned setup?
Yes you and Crofter are right, 50 cells per week is a lot of work. I don't use an incubator and go directly into nucleus colonies with the cells. I did this last year for 6 weeks in a row and made close to 300 nucleus colonies I can say it was a ton of work however like most bee work a whole lot of enjoyment and satisfaction is well.
Making up those nucleus colonies was very labour intensive. Sometimes I broke up full size colonies to make the nuc colonies but I also had some brood factories which I liked much better. One of the problems with breaking up full size colonies is if the queen can't be found one of the cells is automatically sacrificed and breaking up a colony is a chaotic situation.
I am planning to have about 80 brood factories this year, for me that is a much saner way to go.
A question on the cell builder you described above, what is the top box that you shake the bees into? Do you start with a double deep colony and then separate them with the double screen board keeping the queen in the bottom? Or do you add a box on top with combs, brood, food etc and then shake the bees into that?
With your system how do you feel about the cell quality?