Non-American here, doesn't the US have any tv programmes that are shown every year on the 31st?
Here in NL since the 50s/60s we have at least one comedian doing a 1-1.5 hour routine focused on the last year. These shows are shown on television and are usually the biggest hit this evening, at least 50% of the country watches.
Most channels have some kind of news recap, or an In Memoriam tv show looking back at all celebrities/important people who died this year.
I know in Germany and some other European countries, at least one channel shows a comedy sketch from 1963: Dinner for One.
And for the 1st of January I am going to throw in the Vienna New Year's concert. Also shown each year throughout Europe.