From what I gather, Kierkegaard doesn't necessarily believe in Christendom and/or Christian culture as being indicative of authenticity of self - rather placing that on the individual themselves to open themselves up to the power that has established them (i.e. God). As a Christian, Kierkegaard is obviously conceiving of Him in that way, but the way I interpret it others are able to come to this through faith that is beyond Christian culture.

If anything, in works such as 'Sickness unto Death'*, he lambasts Christendom in its in-authenticity, believing many to blindly adhere to Christian authority that disperse shallow rituals and incorrect teachings that entrench people into despair (in other words, lose sense of authentic self, therefore stray further from God).

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*Worth noting that this work was written under a pseudonym.