Is it Time for an Employee to Go?

I am currently a Manager in IT Operations and my team is responsible for end-user support, we are the single point of contact to IT for service disruptions and other requests. I have a difficult employee and I'm wondering if it is time for him to go to improve the team overall.

This individual consistenly scores high on our objective measures (i.e. Average resolution time, customer survey scores, time tracking, etc.), however his behaviors are poor. During team meetings if he disagree's with something it just says "This is stupid". I require that all team members follow a specific procedure when we have Major Issues. This tech does not believe its effecient and simply does not do it, this is one of example of many.

Some things I've done wrong. The majority time I've only provide feedback in quarterly reviews and some during 1-on-1's. I've tried to provide regular positive or negative feedback in the past, but the employee gets very defensive on even the lightest piece of negative feedback. When I give positive feedback, the person seems completey unimpacted by this. I'll see where the person begins doing better in an area (i.e. taking good notes) so I will give positive feedback to encourage. Then a week later I see that he has returned to the previous behavior, and from there its a ping pong match. This has made me more tenative in giving feedback because of the defensive behavior.

The tech complains to the team a great deal about the company and about procedures I've implemented, these are not brought up to me, just complaining to the team. Other teams do not see this person as easy to work with. 

I've also tried giving this person increased responsibility to further challenge him. He completes these tasks on time, but only precisely what I've assigned. There is no initiative taken to overcome hurdles or "think outside the box".

Overall, I feel like I've failed the employee because I have not given regular feedback using the feedback model. I've been managing the person for two years and we are at a pivotal point of change. I do not see the person jumping on board with the change, and that combined with attitude, I'm wondering if it is time to let the person go? The other part is, if so, how to go about?