Mathematicians arent always supposed to solve "real world" problems, but often enough to provide ways for others to solve real world problems. The question he "solves" in here might be relavant for someone, e.g. a physicist, using Grandi's Series only to end up with a singularity having 0, 1 and 1/2 as possible results. And maybe tell him to just follow a different approach, if he is looking for specific results.

Personally, the video actually answered how Haruki Murakamis novel "Hard-Boiled Wonderland and the End of the World" ends.

For your question why Dr. James Grime makes "senseless" Youtube-vids https://www.youtube.com/watch?v=Yexc19j3TjE

Almost every Numberphile video, same as Computerphile, 60 Symbols and others, isnt made for the purpose of solving complex scientific issues, but to show what the discipline is/can be about.

It works as a PR and recruitment tool, to show what a discipline has to offer and often enough just to give insights to subject foreign people.

A lot of Academic work force nowadays is actually just to get sponsors for ur research (one of my sociology professors estimated up to 1/3), therefore raising people's awareness for the importance of science, is a a good way to get finacial ressources in public founded academic tradition.