Him not coming out for the lap of appreciation is massive. Either he has made the decision, in which case it's a major dereliction of duty as Everton Manager. I would imagine it's been from above though, and somebody has advised him not too. The question then is, if he isn't safe/allowed to walk around the pitch and say thank you to the fans, such is the hostility towards him, how can he be the right person to lead the team forward? This would be a question that has a very obvious answer to almost any business.
In pretty much any other company you witness, or are in contact with or whatever, I would not hesitate to say that's him done as it would be completely illogical to keep him on after that. However knowing the personnel of the Everton board, you can't really rule anything out. They have made many strange, bizarre, ridiculous and insulting decisions in the past so I certainly won't put my reputation on the line to say he's done as it would make me reliant on their actions. That being said, in any decent business, with any semi credible person, an employee who is deemed unable to say thank you and face the paying customers would not be left in a position of central importance for another season.