I’ve been thinking more about this …
It seems to me that it is impossible to design a machine that will produce random results although you could build pseudo random behaviour into it - such as shaking or moving part of the apparatus.
If I were designing the project so that a human dropped the stick (I would only have one) I would instruct her or him to drop it with the greatest repeated accuracy they could achieve and as fast as they could do it. Then there seems a possibility that the inevitable small variations will be random, or at least subconscious.
I think you may be able to apply the same logic to a machine. If you designed an incredibly precise machine and measured the results with extreme precision you would probably find a random variation due to the tiny effects of molecular disturbances. But such a machine would cost £100,000 +.
I suspect the problem with a less precise machine (I.e. an affordable machine) is to distinguish between its normal non-random (but perhaps very confusing) cycle of behaviour and the truly random variations in that cycle. I wonder if you carried out batches of (say) 1000 tests and looked for variations between the batches would that variation be random?
The best I can think of for a simple machine is one which drops the sticks on their end so that they bounce before they land on the stripes. But even then you would need to do a lot of tests to check whether there is a repeating non-random pattern. I’m not sure whether randomness would be improved by using the same stick all the time or several or a large number of sticks.