They are somewhat split, their Bavarian branch is calling themselves "CSU" - "Christian Social Union", yet one more irony.
Also in the recent years, the more open Nazi supporters split off and formed two separate parties, first the AfD, and later, the "Werteunion" - "Values Union". You can guess what their "values" are.
As much as I dislike the CDU and especially their current leadership, there is still a large number of people in there who believe in democratic processes.
Which is why they do support that kind of leadership and tolerate(d) vile individuals like the Guy of the original post and the founders of the AfD and the Werteunion in their mids.
I don't think there's an all-encompassing answer to that. Organizations, parties included, are simply not without their inner contradictions and don't behave as a single organism. We've butted heads on pretty much that issue before. :)
I'll also note that in many cases, the CDU understanding of democracy is maximally representative which I think breeds antidemocratic sentiment. I.e. they want people to vote every four years but that is all they expect voters to decide, and they don't want to be transparent toward voters.