Excellent answer! Makes sense too in that 'grandma' doesn't celebrte birthdays on the 'right' day, so...
I've been thinking a lot about this - for years, actually. I've come to the conclusion that even tho' some of the 'pagan' influences still exist in most people's celebration of Christmas, the purpose in the early church was to place Christ in the midst of any celebration. They didn't do it with evil intent, but to put Jesus in his rightful place smack in the middle of any and all celebrations or rememberances. So it seems they wanted to overcome the pagan influence with Christ, instead of just 'adopting' the same date to celebrate Jesus' birth, they attempted to overshadow paganism with Christianity. It half-way worked. lol. Jesus isn't only the 'reason for the season' - he's the reason for joy EVERY day too. There really is a problem with thinking that everyone ELSE has evil in their hearts concerning religious thought - depressing way to think and live.
Anyway, I liked your answer, PS, and think it's easy enough to understand - even for 'grandma'!