I'm not sure about the other suggestions on that video, but I thought that the explanation why Mozart chose E-flat Major was rather unsatisfactory.
In music we have 24 keys. If all keys were used equally, then we would have a probability of 1/24 = 4.2% that a classical composer would choose that key.
When you consider that more compositions are written in keys with fewer accidentals than keys with 6 or 7 flats or sharps, then the probability that a classical composer would choose E-flat major (that contains only 3 flats) goes up to perhaps 5 or 6%.
imslp.org – a well-known repository of public domain compositions – lists 741 compositions by Mozart, 47 of which are in E-flat major, which is 6.3% – well within the expected range.
In my opinion, it is more likely that Mozart chose E-flat major because the traditional character of that key is love, devotion, intimacy, and conversation with God.
Another point was made on the hidden meaning of transposing the middle C one octave up in the 3rd chord, which also sounded like a bit of malarky to me. That technique is called position change and has been used extensively in music compositions for centuries. It can evoke an emotional response, but it is just as often used to open up a register for what's coming next.