The fact that all countable subsets of the real numbers have Lebesgue measure zero (as proved in the video) implies that the real numbers are uncountable. Otherwise the real numbers would have Lebesgue measure zero, which is absurd.

This approach to proving R uncountable is carried out more formally here (proof A.3): https://link.springer.com/content/pdf/bbm%3A978-1-4614-8854-...

Totally agree! When I was in school, I took a logic class that used Boolos & Jeffrey's "Computability and Logic", which is a really great book. Chapter 2 is a very clear presentation of Cantor's diagonalization proof that there exist uncountable sets. It blew my mind.