There is one good reason why C tends to be taught first and only later C++ (if ever or as an elective). C is usually part of the first introductory programming courses, where the students need to learn some basic algorithmic thinking first, with many having never programmed before. While C is far from an ideal language for that, the idea is, especially in engineering and not CS oriented curricula, that the student is given a tool they can use to solve real world (as opposed to toy) problems right away.
The CS students will have opportunities to learn other languages than their Python or Scheme or whatever is used in the 1st semester programming course later. For the engineering students these courses could be the only formal exposure to programming they will get during their study, that's why the language of choice is often C.
C++ in this context would be unnecessarily complex - I can't talk about OOP or metaprogramming or iterators to people who don't have a clue what a loop is or how to represent data using variables yet. You could certainly do this using C++ syntax as well, but then there would be a lot of things you would need to handwave away as something the students should take for granted and not focus on just yet (e.g. C++ strings or STL vectors and such). That's a bad thing pedagogically, because such "black magic" will reliably confuse the students. The reasons for the choice are really not technical but pedagogical in this case. (been there, done that, having taught undergraduate programming courses).
On the other hand, where the students know to program already because they are moving from e.g. Java, C# or something else, then yes, start with C++ right away. At that point the foundations should be fairly solid already. I believe that that is the situation the video is assuming.
BTW, someone said above that C++ is superset of C - it isn't. At least not a proper superset. There is plenty of perfectly valid mainstream (no obscure tricks) C code that will not compile using C++ compiler. For example C99 style struct initialization doesn't work in C++, there are more reserved words in C++ than in C (new, delete, template, auto has different meaning, etc.)