Maddening mess? Come on, this is silly nitpicking. Airport codes are as clean an logical as it gets with century-old international standards. In fact, most of "maddening mess" in the video can be essentially ignored, which author basically admits by stating closer to the end that these are used by different organizations for different purposes. It is just that story is being told backwards and in a snarky voice for dramatic effect, otherwise it would become clear that IATA codes are pretty much the most natural and logical thing in the world, which would rob author of a passable video topic. There is no real pressure to use ICAO codes instead of IATA codes, and the switch would be quite complicated because of all the legacy in GDS systems, not to mention organizational effort.
From the top of my head, I can come up with only one reasonably annoying thing about IATA airport codes, which is much less than most other legacy standards. While EWR can be used both as a city code and as an airport code, according to every airline in the world airport EWR belonged to the city NYC until this October (because it's "New York metropolitan area", despite being in a different state). This is no longer true, but there still are a couple of other peculiar cases like that in other countries. This is important, because airlines have different rules for fares, exchange rules and stuff for airports in the same city and different cities (like there will be different fares for round-trips and non-round-trips, and the definition of round-trip obviously depends on different airports being in the same city or not).
From the top of my head, I can come up with only one reasonably annoying thing about IATA airport codes, which is much less than most other legacy standards. While EWR can be used both as a city code and as an airport code, according to every airline in the world airport EWR belonged to the city NYC until this October (because it's "New York metropolitan area", despite being in a different state). This is no longer true, but there still are a couple of other peculiar cases like that in other countries. This is important, because airlines have different rules for fares, exchange rules and stuff for airports in the same city and different cities (like there will be different fares for round-trips and non-round-trips, and the definition of round-trip obviously depends on different airports being in the same city or not).