What this probably means is that the anti-skate set by your tonearm is inaccurately calibrated; if it sounds better without the weight it is probably adding too great an anti-skate force. The only way to correct for that is to not rely on the the calibration, but on the effect it has. That means either an ungrooved disc (most say this is unreliable) or something like a Wally Skater.
Are those who always advocate using no anti-skate already biased?
I would think they are, by definition, unbiased!

