I suppose I ought to watch the whole video, but gave up when he said B equals H. They are not the same thing. The unit H is the magnetic field strength in amps per metre, the unit B is the magnetic flux density in teslas. They are proportional in a non-magnetic medium like air. But as soon as a ferromagnetic material is introduced the relationship becomes non-linear and complex, in a mathematical sense, ie, complex numbers.
He is right that pure iron has a higher permeability than carbon steel. If I recall my lectures the problem with carbon steel is not the carbon per se, although it is diamagnetic, but when the carbon precipitates out in the form of carbides.
As hinted at by MG there are specific steels intended for magnetic applications, known as electrical steels. In simple terms the carbon is replaced by a few percent of silicon. Permeability is almost as good as pure iron, BH characteristics can be tailored and it has a much higher electrical resistance than iron. This reduces losses due to eddy currents.
Andrew