The video is should't say the map is a lie or wrong, that's not true, it should explain how a projection works, so people can really know what are they looking at instead of go around saying maps are not accurate... Mercator projection was originally made for sailing so it preserves angles, you can and should trust it if you know how to read it... if you want the map for something else i.e. comparing sizes, you should simple use other projection...

Schools should really take a look into this matter, at least show graphically how a projection is done, it's no so hard and it should avoid things like that video that will make a lot of people go trough life saying maps are not useful...