The claim that Christmas is Saturnalia renamed is repeated everywhere and does not survive contact with the Roman calendar.
Saturnalia was 17 December, extended in the imperial period to the 17th through the 23rd. It is not 25 December. Its characteristic elements, the reversal of master and slave, the licence, the gambling, the mock king, have no counterpart in the Christian feast, and the gift-giving associated with it belonged to the Sigillaria at the end of the run. Ancient Christian writers who wanted to attack pagan festivals were entirely willing to name them, and none of them says the nativity feast replaced Saturnalia.
The Sol Invictus argument is stronger because the date matches, but it too has a difficulty: the festival of the Unconquered Sun on 25 December is first attested in the very document that first attests the Christian feast, so the claim that Christianity took over an established solar holiday rests on evidence no earlier than the thing it is meant to explain. Aurelian's promotion of the cult in 274 is real; that he fixed a 25 December festival is an inference.
Some later Christian writers, from the fourth century onward, do draw the solar comparison, calling Christ the true sun. That is theological appropriation of imagery, and it is evidence that the coincidence was noticed. It is not evidence about origins.
The evergreens, the tree, the log and the wreath are a separate matter again, largely medieval and early modern and mostly northern European.