Neither of those are necessarily simple. The things you look down on will come back and bite you badly.
The guitarist David Grier will ask players who come to him for advice on complicated tunes to play Happy Birthday. Then when they inevitably fail he will ask "How are you going to play that complicated stuff if you can't play Happy Birthday"? It is kind of a trick because Happy Birthday is very familiar to us so we think it is simple. But there are two or three wide interval jumps in the tune that make it hard to catch by ear. Dixie and Old Folks at Home are the same way. Most people will fail on the first try, or even the second or third. Of course you do need to be able to play it in all 12 keys also
Christmas carols are all over the map on difficulty. Try Sleigh Ride, the entire song, not just the first part. It has three parts, changes keys several times, between and within parts, and in some versions modulates up half a step before going another round. Not simple, not even remotely. The Christmas Song (Chestnuts Roasting on an Open Fire) also changes keys several times particularly within the bridge. I would not recommend starting with those. Even the simple songs can be tricky, particularly to play them perfectly and in time. Jingle Bells repeats the same note over and over and over on the chorus. Some people have difficulty and lose track of where they are. Joy to the World has a first line that is very simple but it has to be timed right, If the timing is wrong it is another tune. Even very traditional carols like O Holy Night or We Three Kings have modulations in them where the key shifts for a bit. White Christmas has a lot of chromatic behavior in the melody. The melodies are not necessarily simple.
Of course the point of playing simple songs is to get your mechanics right. You want every note to ring out perfectly, no misses, no buzzes, no squeaks, everything in time. And for that you do not want a challenging melody where even finding the right notes is a problem.
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