Why would a climax community such as oak woodland have a higher species diversity?
Because of order and stability.
I understand a climax community as a gathering of populations which yields an overall equilibrum, stability system, with hardly no changes over time.
Thus, organims in there must be adapted very well, interacting with other species positively (there is predation of course).
In my opinion, Oak wookland has not suffered the introduction of invasive species or any significant climate change phenomenon.
So I'd say that equilibrum is everything needed by a system to function effectively.