I think that it is wrong to see Christmas as only a celebration of the birth of Christ. If that was the purpose, we would celebrate it closer to the time of Christ's birth, not at the start of winter.
Almost every non-equitorial culture on earth has some sort of major celebration near or on the winter solstice. It is my understanding that the timing of Christmas for celebrating the birth of Christ was adopted to fit well with the other solstice celebrations in Rome at that time. It has always been a hybrid of Christian and secular traditions.
To follow the original poster's logic to an extreme, we should also avoid non-religious carols like "Jingle Bells" or "White Christmas". We should avoid non-religious decorations like trees and lights. We should also avoid Santa Claus, Frosty and Rudolph.
My view is that if you want to restrict Christmas to only it's "true" meaning, you should move it to a different time of year and keep it focused only on Christ. Personally, I love the mixture of religious symbolism and secular jocularity. It's a delightful holiday season with both religious and secular traditions. If you want to send me cards showing baby Jesus, feel free. If you want to send me cards show you and baby Jose, that's great to. Enjoy the holiday season. Celebrate again the birth of Christ, the lengthening of the days, or whatever story of regeneration and new hope that makes you happy.
Just don't go all puritan on me and insist that I only celebrate this time of year the way you decry. It isn't just your party and you didn't even start it.