Yes, it's classic that if you've got a bad AMM, running the engine at idle with the AMM disconnected and getting a good idle is the usual way to show it's a bad AMM. Just realize that the car still won't run well at speed without a good AMM connected.

But you wrote, "...and after a few minutes we put the connector back on and the car started. In order for the car to run we have to go through this process every time..." So I have to caution that you have to shut off the engine every time you connect and disconnect the AMM -- in other words, never disconnect or reconnect it while the engine is on.

A lot of us have come to accept that the main reason the AMM's die is that the thermostat inside the air filter box fails and overheats the AMM with air from the preheat duct (from the exhaust manifold). If you want to protect your next AMM, remove the thermostat from the air filter box, and block off or jam the flapper so that heated air never gets into the box. That thermostat that drives the flapper is just too unreliable.

Good luck.