I had an EML light come on recently,and the code I got suggested a problem with the immobiliser,but the car was starting and running ok.
I have a couple of codes which keep reappearing but they are nothing important so I usually just reset them,one of them is AC related (AC does not work anyway LOL,rarely a problem in scotland

),but a couple of weeks ago I decided to check the AC fuses.Before I checked the fuses I ran a finger down all the fuses to check they were fully engaged - one of them was not fully engaged and I pushed it fully home.
The EML light went out without me resetting it LOL and (so far) has not reappeared.
I am not saying this is the cause of your problem but perhaps worth keeping in mind as an occasional check.