This xkcd "What If?" reminded me of a short story I read, maybe in a compilation of Golden Age SF stories by the likes of Henry Kuttner and Fritz Leiber. It was just a few pages long. A group of rich old geezers are passengers on one of the first faster-than-light ships to leave Earth. The last few sentences of the story describe them gathering around a TV set to watch TV signals that left Earth in the '50s, which the ship has caught up with---it's an episode of "The Howdy Doody Show" that the old guys remember from childhood but thought they would never see again (since it was from the era of live broadcasting before videotape).
And, of course, after I post a bounty, I find some references. It seems this is likely "What Time is It?" by Jack Haldeman II.
There’s a science fiction story that postulates that the first use of faster than light travel is to carry aging baby boomers far enough out into space to see “Howdy Doody” and other shows that were broadcast live and not preserved: “What Time is It?” by Jack Haldeman II (Joe’s brother).
There's a story called "What Time Is It?" by Jack Haldeman https://www.isfdb.org/cgi-bin/title.cgi?53469 - FTL is used to go far enough away from Earth so that aging Baby Boomers can see episodes of Howdy Doody again.
And user14111 found a link to an archive.org copy of Vertex Volume 2 Number 6 where the story appears:
The men were crowded around a large wooden box. It had a three inch screen with a magnifying glass in front of it. They were watching the fuzzy television images that had left the Earth years and years ago— long before video tape had been around to preserve the signals. Some of the men clutched old television schedules; some wore hats with ears on them. A freckled-faced puppet was dancing on the screen while a clown squirted seltzer water.
All together now.
Hey kids! What time is it?