Lots of learning opportunities here. First off, CB350 carbs are known to be a PIA to get right and older bikes with stale fuel are a real problem to keep in shape.
OK. Let's start with the theory: Fuel feed from the tank to the carbs is gravity as Trials mentioned. So that fills the float bowls and teh jets sit in a pool of fuel. As air passes through the carbs, it speeds up and that causes a lower pressure in the venturi compared to the atmospheric pressure in the float bowl and that pressure difference causes fuel to come out of the jets into the airstream. Some call that a vacuum and others say it's the pressure in the float chamber and it's both. It's the difference in pressure that does the work.
Others will tell you that the carb needs a restriction on the inlet to allow the motor to suck fuel out of the jets but that's not really what's going on. For sure, a lack of filters or restrictive filters will potentially change flow a little and cheap pods usually restrict airflow enough to kill performance.
I have run CV crabs with stacks, and foam filters and super short drag race bellmouths and they all can be made to work with a little work. Nothing much fixes cheap pods though.
Your problem could be any number of things. So start at the beginning. New fuel tap may actually be the problem. Remove the fuel tubes from the carbs and put them into a container and turn on the fuel tap. It should flow freely but will usually only flow out of one tube at a time. If it does that, pinch the line that's flowing and if the other side starts to flow it's good.
That happens, I believe because as the flow starts in one side, it creates a low pressure on the other outlet and pulls air in instead of letting fuel out. That's my theory and I'm sticking to it - at least until I see a better explanation.
If the petcock is OK, the next place to check is the float bowl and the float valves to see if they are flowing. Remove one bowl and turn on the fuel and see if it comes out of the float valve. BTW, do this outside and with a container to catch the fuel. repeat on the other side. Fuel doesn't come out as fast as you might expect but as long as its flowing freely, you are OK.
Try that an report back please.