Yes, there is more to dealing with the average seemingly-small sax problem than most people realise. Especially in an ideal world, and when time/skill/equipment are in abundance.
I agree with 1,2,3,4. However, of the tone hole looks reasonably level, and the pad looks to be reasonably flat and in good condition, then it is doubtful IMO that you need to do the rest in most cases. Just re-align the key cup so that the pad closes all around. Put a pad slick (or even some carboard) under the part of the key cup that closes first, and press down on the part that was closing last.
However I reiterate that this is not the cause of any problem with your E. If E and Eb are difficult, then there is a leak further up your sax.