It is not that the birth of Christ would have lost its specialness, rather it would be too-often repetition that would diminish our perception of its specialness.
Just like Christmas, our bodies are a gift most sacred and special. Many consider the commandment "thou shalt not commit adultery" and similar Christian teachings to be too difficult, too restrictive, or simply not very important. But the more we share our bodies, the less sacred they are perceived to be. Even the most vile human traffickers place a premium on those who have not yet shared their bodies with anyone. Even the devil recognizes how invaluable our bodies are. That is why he tries so hard to get us to view others' bodies inappropriately or share our bodies with others. He wants us to lose our sense of reverence for our body and the intimacy that should be held sacred between husband and wife alone.
Just as I should reverence Christ, I should reverence the body God has given me. No person should force indecent exposure. No media outlet should encourage the devaluation of modesty and decency. No government should force exposure of my body, or require physical touching of my body without reasonably strong evidence of endangerment to others.
Any encroachment on these principles is a blatant denial of the God-ordained right of all men and women to have their bodies kept sacred and have them treated with the reverence that true Christians would afford a temple.