George Martin did not write their chord progessions when it came to straight rock. Except only later if it involved string arrangements and even their espeically McCartney would hum the parts. You don't need to know how to read music to write complex chord progressions.. They may not have been able to read or write music but that doesn't mean they didn't understand it, the same way someone can speak english without reading or writing it. Remember, they had the best education of all: learning and studying 100's of songs during the Hamburg period. Their ability with intersting chord changes came from emulating jazz standards like "Till There Was You". The use of modal fourths and fifths comes from their skiffle background which is folk influenced .
They were naturally gifted, came up with melodies, rhythms, harmonies, chords to fit their ideas, and played what made sense to them in terms of what they were trying to accomplish. When experts marvel at how a song can contains passages in 4/4, 3/4, 5/4, etc., and "strange" modes like the mixolydian, it's all after the fact. I don't think they were aware of any of this and just went with what felt right.
Have you ever seen a note-for-note transcription of a Jimi Hendrix solo? It's like, who could possibly think of this? Well, he didn't. He just played, and someone else later figured out how to put it on paper and took note of its significance in terms of music theory.
Brian Wilson who also could not read or write music.