That's indeed a very likely reason why it is celebrated when it is, but... Christmas is/was initially supposed to be the celebration of the arrival of the Christ child or Emmanuel (which means God with us). It's initial intended purpose is probably best summed up as the celebration of the fulfillment of Mathew 1:21 KJV... "And she shall bring forth a son, and thou shalt call his name JESUS: for he shall save his people from their sins." Or John 3:16 NKJV... "For God so loved the world that He gave His only begotten Son, that whoever believes in Him should not perish but have everlasting life."
However, again, you are likely correct that the Winter Solstice is probably why the date for the Christmas celebration was chosen, since Christmas was most likely not celebrated until after Constantine became the Roman Emperor around 325 AD or so. Also, it is correct that Christmas has most definitely taken on many meanings with many different people since then.... Giving of presents, Santa Claus, Christmas decorations, Being kind & good to others, time for family reunions, etc. But, again, the original intended meaning was the celebration of the birth of Jesus & that's why I celebrate Christmas, but I know that others have their own reasons & I have to respect that.
With that said... I want to wish everyone a Merry Christmas & a happy new year!