Layered sheet metal is not a very cheap, or easy way to make a sprocket. Or much of anything. They use the technique to make transformers, but for totally different reasons.
I would be slicing the final thickness you want off of round bar, with a small 4x6 horizontal bandsaw- its gonna be cheaper and quicker in the long run, and the bandsaw is good for a million other things.
Yes, you could cut circles in 16ga with a flycutter- but to do it in much quantity, you would probably want a milling machine, with a collet system that actually locks in place with a drawbar- nothing quite as exciting as when you flycutter overpowers the friction fit of your morse taper drill chuck, and decides to go for a spin around the shop.
2" circles are usually stamped on a punch press. And you can buy em cheaply from companies that do this by the thousand. Like Wagner-
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The tonnage required to punch a 2" circle from 16 ga is more than you would think- probably around 12 tons. Can you hit a hammer that hard?