There is not a difference between the two situations in the way you’re claiming, and indeed the differentiation point of view is used fruitfully on both factory floors and in more complex convex optimization problems. For example, see the connection between dual variables and their indication of how slack or taught constraints are in convex optimization, and how this can be interpreted as a relative tradeoff price between each of the constrained resources.
In your factory floor example, the constraints would be the throughput of each machine, and (assuming you’re trying to maximize the throughput of the entire process), the dual variables would be zero everywhere except at that machine where it is the negative derivative of the throughput of the entire process with respect to the throughput of the constraining machine, and we could determine indeed the tight constraint is the throughput of the relevant machine by looking at the derivative which is significantly greater than all others.
There is not a difference between the two situations in the way you’re claiming, and indeed the differentiation point of view is used fruitfully on both factory floors and in more complex convex optimization problems. For example, see the connection between dual variables and their indication of how slack or taught constraints are in convex optimization, and how this can be interpreted as a relative tradeoff price between each of the constrained resources.
In your factory floor example, the constraints would be the throughput of each machine, and (assuming you’re trying to maximize the throughput of the entire process), the dual variables would be zero everywhere except at that machine where it is the negative derivative of the throughput of the entire process with respect to the throughput of the constraining machine, and we could determine indeed the tight constraint is the throughput of the relevant machine by looking at the derivative which is significantly greater than all others.
Practical problems also often have a similar sparse structure to their constraining inputs too, but just because not every constraint is exactly zero except one doesn’t mean those non-zero constraints are secretly not actually constraining, or its unprincipled to use the same math or intuitions to reason about both situations.