Hi,
I had the similar issue, accessories working, but the engine not turning over until you take the key out and press the remote locking on then off. i thought it was an immobiliser issue.
After a while I found it was simply that the two metal tabs that the key touches when the key is in the ignition weren't making contact with the key, pressed the key inwards when turning it around to start and hey presto! starts first-time/everytime!
hope that helps