That's a really good point and absolutely true. And there is more to it than just the choice of lowest note. How that note is executed is often more important than just the pitch class. Take the prelude to Bach Suite 1. In the first major key section (before it goes around the houses), the first note of every other bar is marked Tenuto. It gets more time than the 1/16 might suggest. In the first 2 bars, low note (G) it is Root. In bar 3 it is m3 of an Em chord but still gets the Tenuto to maintain the key centre. Here's a pic of my annotated copy showing how I interpret the Tenuto...
It's interesting for purposes of this discussion to note that, in the first four bars of this example by Bach, the low G played on the downbeat of each measure is not necessarily the root of the chord. In bars 1 and 4, the harmony is G and the low G is indeed the root. But in bar 2, the harmony is C/G with the 5th in the bass. And in bar 3, the harmony is D7/G. Classical theory textbooks call this technique "pedal point."
The answer to the question, do you always need to play the root as the lowest note on the downbeat, would seem to be no, at least in the musical language of Bach.