This might sound like a stupid question, but when you rotate the trim pot clockwise, does that effectively reduce the hum, or is the hum reduced by turning it counter-clockwise?
Both, really. You're adjusting the amount of out-of-phase hum that's added to the signal. There's a sweet spot where it will cancel most of the hum from a given pickup- to little or too much and you'll introduce more hum.
Adjusting the trim pot only removes the low frequency hum and has no other tonal affects that I can hear. Same for removing the battery - all that changes is the noise level but otherwise everything sounds identical.
The Silent Circuit was designed not to affect the tone of the pickup. Removing it from the circuit should not make any difference.