Why can't a control unit detect this 12 volt signal while my multimeter can?
I would suspect that the LED sign is using a DC-DC converter internally and that there is no connection between the power in and the power out. I've represented this as a transformer isolation in Figure 1.







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Figure 1. This circuit has no return path for the stop button current. The LED won't light.

The result is that when you feed +12 V in to the new sign that the LED won't light as there is no current return path.







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Figure 2. By adding a relay we can create a complete circuit for the STOP button and for the LED circuit.

Adding the relay solves this problem but you must be connecting the LED to the sign's internal supply.







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Figure 3. Working without a relay.

Connecting A-A or B-B (but not both) in Figure 3 would allow the circuit to work without a relay as there would be a reference between the two circuits. (You would need to reverse-engineer the LED sign to be sure. In particular, you would need to confirm that the LED sign is 12 V internally.) The downside is that the manufacturer could change the device design at some stage and you would risk damage on a future replacement.