I've noticed a lot of songs that start with the sixth and then go to the fourth. Starting with the sixth isn't a way to set the key as far as I know. What makes this progression popular?
They aren't actually beginning on the vi chord of a major key, but rather on the i chord of a minor key. Choosing a i minor chord that is also the vi chord of a major key makes writing pop much easier, since it is simply the relative minor of that key, which makes it extremely easy to modulate to the major key. For example, in the key of C, C is the I chord and Am is the vi chord. But starting on the Am and using the chords from C (which are the same as the chords of Am) simply places the song into the key of Am. The classic chord progression based on the beginning that you mentioned in your question is the i-VI-III-VII progression (Am-F-C-G), which is also vi-IV-I-V (Am-F-C-G), if you wanted to write it based on C (which you probably wouldn't). What makes the progression popular in pop music is that it resolves very nicely and uses only 'bland' chords from the key with no dissonance or added intervals and is therefore very pleasant/inoffensive to the ear. The more simple way of saying that is that "it just sounds good". In addition, because all the chords are in the key, pop singers can simply sing in the pentatonic scale or natural minor scale, which is extremely common in pop music.
I think the better question here is why so many pop songs nowadays (and for a long time earlier) use the vi, IV, I, and V chords, but since that question's been done to death on SE, my take is that there probably isn't a huge statistical correlation showing growing frequency of porgressions increasing over time, but rather that once one trend of chords (say, starting on the IV) gets hackneyed, the songwriters in the industry mix it up, which starts a trend on the next chord. Another alternate idea is that if you haven't been observing the first chord in every popular music tune you hear, it could be that when you notice the vi chord, you remember it later, and when you don't, you tend to forget the data that don't fit the trend, skewing the data that you remember to the trend.