I think there's definitely a 'fear of change' with some fans.
We spent so many years chopping and changing managers only to become further destabilised and drift closer to what seemed like an inevitable relegation at some points, that having a manager who has brought some stability and a chance for us fans to catch out collected breath, has brought some much needed respite.
I was very skeptical when he was brought in, but have been pleased with what he's done so far. Yes, some of the old Moyes-ism's remain (leaving subs too late, seemingly not giving younger players a fair shake etc.) but my personal metric for Everton is season by season improvement. Slowey & surely if needed. He has done that so far.
Is he the manager to plan out our long term vision? No. But even I admit to having a slight feeling of dread at changing yet another manager. It's the Everton PTSD if you will, and yes, I know that's not how we should be basing our decision on whether or not to bring in a fresh face!
Barring a complete capitulation between now and the end of the season, I think he should be given the next season to continue setting the foundation/bedrock of a competitive team and we should definitely be looking at/courting a long term replacement to step in after that.