Just an update... I spoke to the owner of the 2nd company (not the original installer). He was able to find out where the first guy bought the system and they are replacing the entire valve head/assembly under warranty on Tuesday. He thinks the valve is defective and that's why it isn't drawing any water from the brine tank.
So, at least I effectively cut out the first guy. Mission accomplished there.
And if for whatever reason it still isn't drawing water after the new valve goes on, then a whole bunch of possibilities are eliminated since it's a brand new valve.
Will let you guys know what happens. In two days. Wish me luck.
I would put enough water in to put the water level over the air check valve, and see if the system draws water then.
Water softener is fixed! Second company sent their lead technician back and he replaced the valve head with a new one under warranty. It's another 5 button display, but it seems like an upgraded model compared to what I had before. The new display is backlit and has way more options. It even has a mode where it tells you how many gallons per minute are being used when water is running. The technician set hardness at 10 GPG, fill at 10.5 lbs, capacity at 32 x 1,000 (which equals 3,200 gallons), and automatic 14 day regen if capacity isn't used by then. He also set the cycles as follows: 1 - Backwash @ 10 mins, 2 - Brine DN @ 60 mins, 3 - Rinse @ 7 mins, and 4 - Fill @ 6:57. He was also super thorough and pulled out the entire check valve rod out of the brine well. He said one of the nuts didn't seem like it was threaded correctly and he re-threaded it to make sure there was no air leak.
Once he installed the new valve he immediately said the suction was way stronger on the Brine Draw cycle And sure enough, it sucked water out of the brine tank very quickly. I added three new 40lb bags of salt crystals to the brine tank, let it soak for 3 hours, and then ran a Regeneration.
Once the 60 minute cycle started, it took less than 9 minutes to draw the brine out of the tank. The water level was lowered well below the salt grid, and looking into the brine well the water level was essentially at the bottom part of the check valve. The Fill cycle ran and put about the same amount of water back into the brine tank - I'm guessing it's approximately 3.5 gallons.
All and all, I am super pumped that the system is finally working the way it's supposed to work. Thank you Reach 4 for all of your responses!