There is a lot going on here.
First, ignore your family room. The 4 speakers are likely setup to try to be the rooms left and right and surround left & right speakers for a 'surround' setup. Builders almost never know what the heck they are doing, so, they are likely the wrong speakers in the wrong location and not setup right at all. But, at least each speaker was individually run.
The rest of the house is still a HUGE question mark. I disagree that the Sony can't power them. Or, I partially disagree. The speakers should all sound the same. That is, they would all be quiet if the Sony didn't have enough power for one pair of speakers, it wouldn't have enough power for any of them.
So, it sounds like the wiring is SUPER wonky.
Three pairs of speakers (6 speakers) requires 12 wires to run them. The question becomes, how are things actually wired? Why aren't there 12 wires in your family room? One pair for each of the speakers? What is going on with the wiring such that it is so messed up?
My guess is that they ran four wires to the first volume control, then four wires to the next volume control, and so on. That they completely messed up the wiring.
If you turn up the volume all the way in the quieter areas, does adjusting the volume in the first area impact the volume in the other areas?
I'm not even sure of a good way to correct this, but I would think that you could rewire things, once you know what is going on behind the volume controls and I expect that behind the volume controls instead of four wires coming in (power from the amp) and four wires going up to speakers in the room, there is ALSO another set of four wires which is going on to another volume control.
If that's the case, how things are wired starts getting wonky and you will want to take photos and post them and describe what is going on behind the volume controls.
In a proper system, if you remove one volume control, it should have no impact on other rooms with other volume controls.
You shouldn't need to remove any speakers, but it really wouldn't hurt considering how screwy things are, just to make sure there is only one pair of wires going into each speaker and there aren't any stray wires connected up to the speakers as well.