word choice - Determining which good sentiment to wish at each holiday

There's no rule; they're all idiomatic. Even though 'merry', 'happy', 'jolly', and synonyms are, well, synonymous, it doesn't make them interchangeable. People just use patterns over and over again to the point where the minutest change sounds strange (this should be part of the definition of 'idiom'). Sometimes such patterns arise 'organically' (Out of the common ether), and other times one person says something clever, in person or in media, and people start copying it.

As to alternatives for Christmas and other festive holidays, one could use good, great, or wonderful. They are both innocuous and non-idiomatic enough to not sound too strange:

have a good Christmas

have a great Thanksgiving

have a wonderful Easter holiday

As to holidays in general, correct, not every holiday warrants either a 'happy' greeting or even a special greeting at all, and only culture will tell you what is appropriate. Many religious and political holidays (Good Friday/Tisha b'Av/Ashura or Memorial Day) you just show up and commiserate.

Of course, by cultural analogy with the juggernaut of Christmas, many 'special' days seem to warrant a special similar greeting (like Happy Halloween where historically there never was one). There's no necessity here at all, it's just what people like to do and like to copy in others.