Yeah, it is a bit hard to explain.
With the original, simpler design you have only one tube going straight up from the container. It fills up with solution on the bottom, and when the pump kicks in, that solution is pushed into the tank. Now as soon as all the solution has left the tubing, the back pressure against the lower checkvalve becomes zero, and slowly, more solution trickles into the tubing and is pushed out to the tank. When that happens, little droplets remain in the tubing and prevent it from refilling.
So if you restrict the outflow a little bit, all solution is pushed out, but then no more solution enters the tubing since the air pump creates some back pressure against the check valve. Once the pump stops, the back pressure is gone, and the tubing fill again.
The issue was that I couldn't get that to work. Either I restricted the outflow too much, which prevented the refill, or it was too open, which lead to droplets making it through the check valve while the pump was running, and then prevented the refill.
The work around with the DIY rubber glove solenoid makes that work. Back pressure while the pump is working, no back pressure while pump is off.