I think the connected Assembler should pull its needed stuff out of the container, like the robots do in factorio. They do not have to be programmed. They just pick the stuff, which is needed by the factory.
But i understand, that we have a different situation here, as the belts are connected directly to an assembler. Instead of picking stuff from the belt, the belts feed the machinery directly. So, you are right, the container should have a filter function on it's output. Only what you have set in the filter should go out on the belt.
In the actual state your factory will be clogged sooner or later, if you put different items in your container, because everything will go out on that belt.
So actually you have to use a seperate container for each item, which is not very practicable.