We'll never know, will we? None of us were there
Firstly - a huge amount of illustrations show the arrow on the "wrong" side of the bow from the 100 Years War period, so that's either artistic license, ill-informed scribes/artists or an accurate representation of a style we've either forgotten about or didn't realise was in use. If you believe the "every man was to shoot a bow" rule, it's unlikely that the artists were completely foreign to the basics of archery, so would be odd for them to get it wrong so often. Still, could easily be a simple mistake copied by other artists later on.
Secondly - shooting on the right side of the bow over the thumb affects the tension and release of the string. The reason it works for the Japanese archers is because they use a thumb-ring to release, which doesn't cause the string to pull inwards and disrupt the flight of the arrow.
If you try and shoot with a Mediterranean / 3 finger under release AND over the thumb, the fingers will drag the string backwards and twist the string to the right, clockwise. This will roll the arrow away from the side of the bow, either making the flight unpredictable or just causing the arrow to fall off the thumb. With the arrow on the left side of the bow, resting over the knuckle, the clockwise twist of the string pulls the arrow inwards, against the bow. The bow keeps it in place, and the release is consistent and predictable.