This phenomenon of using English words (medium, medium rare) in Russian language nowadays is more related to rebranding/marketing and increased interest amongst Russian youth towards Western/European countries, related either to tourism or wide access of commercial goods (which wasn't available as much until early 2000's) and social media.
Rebranding in a way helps restaurants and cafés to disassociate themselves from the late Soviet and post-soviet history (during 90's), where the service and food quality were far from perfect and often not affordable for many Russians. Hence Russians do not have such a habit of going out on a regular basis as Europeans or Americans have. So by using borrowed English words (even though the same translation exists in Russian) allows to indicate that these places and what's related to them stand out far enough from what people used to remember/know.
Also if you've noticed in your video, the person introduces himself as бренд-шеф, instead of повар-технолог, partially because of a historic connotation. Being a повар until more recent times indicated something unprestigious and associated with schools where they used to study, e.g. ПТУ abbreviated not only as профессионально-техническое училище, but often jokingly translated as "Пришёл Типа Учиться" and more offensive phrases. So even these institutions nowadays are renamed into colleges, culinary schools etc. Moreover, a Russian word училище is not so often used anymore, unless it is been always associated with a prestigious school which managed to keep its brand throughout all these years and stay above these trends, like Суворовское училище, as an example.
On the other side of the reasoning why used borrowed words, Russians were always puzzled and in awe for Western/European cultures (may be, because it was so much prohibited in Soviet times) - to have a piece of European clothing, an American chewing gum or a German badminton set was beyond amazing, so using borrowed words nowadays helps to play with these deep-rooted feelings on older generations and even younger generations look up to Europe/America with interest as a lot of current culture comes from there.
So in a way many think, that the fact that you actually have had all this terminology and dishes long ago won't help to make your place/channel look fashionable and these borrowed words often serve a function of buzzwords.
Even though it needs to be mentioned that the opposite movement also has reappeared in Russia, where restaurants and cafés based on traditional Russian cuisine again gain more and more public interest, so you can find such items as гурьевская каша, похлебка, солянка etc, probably even 10 years ago a given Russian person wouldn't be sure what гурьевская каша is.