Post History

After getting various inputs, I can offer a partial answer to my own question. I'm not a native speaker, so feel free to offer a better one.

[The explanation Duden offers](https://www.duden.de/sprachwissen/sprachratgeber/Die-Weihnachten-oder-das-Weihnachten-Grammatik-zum-Fest) covers most of the bases. Key points:

"Weihnachten" is described as a frozen dative plural (ein erstarrter Dativ Plural) of "die Weihnacht", using a medieval plural form. It's plural because Christmas was conceived as lasting from December 25 to Epiphany (the 12 days of Christmas). 

"Die Form Weihnachten ist standardsprachlich im Allgemeinen als ein Neutrum Singular anzusehen: Es war ein schönes Weihnachten." (The form "Weihnachten" is in general standard usage regarded as a neuter singular.) In effect, it's a singular back-construction from the plural of a feminine singular. But in set phrases like "Frohe Weihnachten", it's always treated as a case-invariant plural. "In bestimmten formelhaften Wendungen, vor allem als Wunschformel zum Weihnachtsfest, ist der Plural allgemeinsprachlich und nicht auf den regionalen Sprachgebrauch begrenzt:" (In certain formalized expressions, especially as a Christmas wish formulation, the plural is in general usage and not limited to regional usage.) 

"All diese Schwankungen im Gebrauch des Artikels, des Numerus und des Genus bei der Festbezeichnung Weihnachten lassen sich sprachhistorisch erklären." (All these variations in the use of the article, the number, and the gender of the designation "Weihnachten" have to be explained by linguistic history.) So ultimately the expression is a sort of living fossil, preserved by continuous usage through the centuries.