The winter solstice is on the 21st of December, not the 25th. The dating of Christmas has nothing to do with it.

Christmas is the date it is because it is 9 months (human gestation period) after the Feast of the Annunciation (25th of March) which gets it's date from an old attributed date of Jesus' death.

In fact, the reason the UK tax date is in March is for the same reason - the new year was the Annunciation. Then 12 days difference due to the Julian/Gregorian shift.

Edit:

Also, the link you provided debunks your own claim-

The popular theory that Christians chose Dec. 25 to co-opt the pagan solstice festival of Sol Invictus is not based on strong evidence but on the margin scribblings of an unnamed Syrian monk in the 12th century.

And once again echoes what I said