AE1's are also prone to cracked tracks on the ciurcuit board that goes over the prism. They dry out and crack for no reason, and this can cause the shutter not to fire. Get a multimeter and check the tracks that run over the prism for continuity - follow the tracks that run over the prism until you find a point you can put a multi meter on - from memory you may need to use the IC pins as a test point on the left side.., and there are points on the right side you can use. When you find a track that is not working run a jumper wire between the 2 points and try the shutter..
It has been 20 years sine I repaired these at Canon, but the worst one I ever saw had to have jumper wires installed before it would work....
Also if you have a second A series that works swap the release magnets over and see if that fixes the faul - cleaning the magnets does not always get them working again..
PS - be careful when taking the top cover off a AE1 - if you break the tungsten wire you may as well throw the camera out, as you can't get replacement wire anymore (and from memory they are a pain to adjust..)