If we are just talking about one location the gravitational pulls of the sun and moon sum up to less when they do not pull in the same direction.Yes, but does that make a difference?
Remember, there are tides on both sides of the Earth. There's a high tide directly below the moon, and another high tide on the opposite side of the Earth from the moon.
I notice this on the tide chart
https://tides4fishing.com/ie/connaught/clifden
where I can see that the highest high tides occur when the moon is New and not so much (but still pretty high) when the moon is Full
I assume the same pattern plays out in other locations around the globe.