Does the time signature dictate what chord pattern you use on the left hand?
For example, in "row your boat" a 6/8 time, I'll repeat the chord twice. So if I'm playing the 3 notes of the chord and play it twice I'll get to 6 notes which matches the time signature.
And in "london bridge" which is a 4/4 time I play the chord using two notes (say the 1 and the 5), and I'll play these twice 1 5 1 5 so they match each beat.
So the chords tones in the left hand essentially match the beat (or go between) the beat. Therefore the time signatures have a direct influence on the chord patterns you play on your left hand. Does this sound right?