Where exactly do you have voltage on the 7-pin connector? When you press on the brakes, does the controller put anything on the brake output? (assuming it's a modern Prodigy controller or similar) Those use a PWM signal so your meter might show anything from 1-10V on the DC scale. And it may also show something like 1-6V on the AC scale.
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My impression is the controller is detecting the trailer brakes by sending a very small current (not a voltage) to the brake pin. If there is nothing connected, the voltage developed by this small test current will go near 12V. If the trailer brakes are connected, the test current sill go through the brake electromagnet that has less than ten Ohm resistance, and develop very little voltage. If the 7-pin connector has voltage on the brake wire, but the trailer brake ground is corroded or disconnected where it splices into the trailer light ground wire, that would also explain your symptoms.
If you have a 5-10 Ohm power resistor in stock, you could clip lead it to the trailer brake and ground pins and simulate the circuit to confirm the controller is working or not. Or as has been noted, try another trailer.