Yes, keep it simple. Stick to cogs.
Much more complex, much less likely to work reliably or even at all. In any case, you already said:
My suggestion would be to go with your cogs idea. Make one of the shafts of the cogs, one near the edge of the frame, pass thru to the rear of the frame where you attach another cog.
Let's assume the cog with its shaft passing through the frame is one of the small cogs, and those cogs have 30 teeth, and the larger cogs have 40 teeth.
Let's also say that the cog on the rear of the frame also has 30 teeth. You drive that with a much larger cog with 120 teeth. One complete revolution of the 120 tooth cog causes 4 revolutions of the 30 tooth cogs, and 3 revolutions of the 40 tooth cogs, which is one complete cycle of the patterns.
You then only need one sensor on the 120 tooth cog to detect the home position. Your motor can drive the 120 tooth cog with a small cog on its shaft.