Wiring would typically be red + /black - (right speaker) white +/green - (left speaker), connected to a volume control INPUT, which repeats the same pattern on the output to connect to in room speakers. The color coding matters. If they didn't follow that standard, then you putting speakers on one end and a amp at the front, with wires you chose, but don't match up to what the volume control used, then you may be connected completely wrong. Hard to know for sure without you pulling the volume control out and seeing exactly how it is wired up and ensuring they are following that standard.
The other thing is that the volume control needs to be turned up or the amplifier won't pass much in the way of audio through it. Likewise, the volume control may or may not be in good operational condition based on the quality of it to begin with.
I don't mess much with volume controls, but they tend to be pretty straightforward.
GAUGE, by the way, is a measurement of the thickness of the wiring. 4 conductors, would be common for distributed audio, but 14 gauge would be more typical for the thickness of that wiring. Maybe 16 gauge. Rarely 12 gauge when it comes to distributed audio. It would never be 4 gauge. That would be enough copper to power a Tesla.
That said, posting photos can help with things and may get you some better answers.
I'm not sure what you are using as an 'amplifier', and that may impact things as well. Maybe.
You also want to think about the end game here. What is it you are hoping to achieve from the different locations for audio. Typically stuff like this is used for background music for your home. Music to put on while you are doing your thing. There are dozens of different ways to connect things from a basic amplifier and speaker distribution hub with a single source like a Wiim or Sonos, to a more advanced setup with individual sources and control, which costs more and requires more sources and a bit more effort. Like using 3 Sonos Amps to run the show.
You should plan on a speaker pair in each room, as that's proper stereo in those rooms and really is the way these systems are designed to work properly. If there are existing speaker holes for in-ceiling or in-wall speakers, it is EXTREMELY unusual for someone to remove them when they leave the home. Simple as that. Even if they repair the holes, it is unusual to remove them in the first place. Like removing a chandelier would be weird.
It's clear that they have a oddly separate setup going on with stuff on the main floor and something separate in the basement, but this may not be that big of a deal overall, it just means that the two systems will stay somewhat separate unless you go with Sonos which can network spaces together nicely.
If you own the place and plan to stay there for years, then spending a bit of money getting this all setup and working well is a nice way to go.
Would be worthwhile to do your homework on the setup of speakers and ask questions if you have them, maybe post some photos as well, so you can get some additional feedback of your options and if there are any obstacles you may want to consider.
Not sure what the 6 speakers are on the main floor either. Are they in different rooms without volume controls? Are they in a single room setup for surround sound? Hard to know, but good to know you got them working.