Christ's humanity-implications?

I find it interesting when Christians insist that human institutions are not part God’s plan for salvation. I’m thinking specifically of the institution of a Church, which is not completely human, but is fully human.

Yet God, of his own loving will, became a human being. God fully enters the human condition in his Incarnation.

Any thought on this inconsistency?

I’m not sure exactly what you mean here.

Poorly worded.
The church is human/divine so the Church is not -merely- human.

771 "The one mediator, Christ, established and ever sustains here on earth his holy Church, the community of faith, hope, and charity, as a visible organization through which he communicates truth and grace to all men."184 The Church is at the same time:

  • a "society structured with hierarchical organs and the mystical body of Christ;
  • the visible society and the spiritual community;
  • the earthly Church and the Church endowed with heavenly riches."185

These dimensions together constitute “one complex reality which comes together from a human and a divine element”:186

The Church is essentially both human and divine, visible but endowed with invisible realities, zealous in action and dedicated to contemplation, present in the world, but as a pilgrim, so constituted that in her the human is directed toward and subordinated to the divine, the visible to the invisible, action to contemplation, and this present world to that city yet to come, the object of our quest.187

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