Bachelor's in engineering physics (condensed matter experimental)/EE specializing in semiconductors here. The explanation starting at 4:00 is very accurate.
When he talks about the electrons "feeling" the neighboring atoms, he's talking specifically about a result that follows from the materials being crystalline, that is, having regular ordered structure. The regular structure gives rise to a periodic potential. You plug that periodic potential into the Schrodinger equation and apply continuity conditions and translational symmetry to the wavefunction. Computing the solutions to the Schrodinger equation with those conditions reveals that there are allowed and disallowed energy levels, and also reveals the relationship between energy and momentum in the crystal lattice. You can step through this by reading the wikipedia page on the Kronig-Penney Model. This depends on the periodicity, which obviously can change depending on direction in a crystal.
His explanation, and the result that "the" band gap is a single number, isn't dishonest because when we grow semiconductor devices, we grow them such that the crystal is oriented such that current flows in the desired direction, so that simple result holds true.
Even his portrayal of the bands leaning down as potential/voltage is applied mirrors how potential change is shown in diagrams of semiconductor devices, see Streetman and Banerjee - Solid State Electronic Devices.