Currently brewing machines like the brewery, distillary and bottler accept fluid input from the funnel from the fluid funnel mod but do not correctly transfer fluid from the output fluid slot to the funnel.

According to the code

There are now two mods that add a block like this: https://minecraft.curseforge.com/projects/fluid-funnel and https://minecraft.curseforge.com/projects/flopper?gameCategorySlug=mc-mods&projectID=290404. They both work pretty much the same way in that they add a low tech, vanilla like block that passively transports liquids from one block to another.

https://imgur.com/a/fXGTH

In the screenshot above, the fluid funnel is the block that looks like a hopper. Up top is a 2x2 infinite water source contained by the stone blocks. This water is funneled to the brewery from the infinite water source above the funnel to the left into the side. Which works fine because like you said, the sides are perfectly valid inputs/outputs. However, because the funnel and flopper both have the input up top and output to the sides or bottom, in the set up here, only water is moved from one block to another. The output fluid from the brewery is not transported and since water isn't a valid input fluid for distillation, the water builds up in both the buffers of the flopper and funnel.

So there is fluid transport, it just isn't the right fluid. If you did this with hoppers and brewing stands/furnaces, the output slot is the only one that can be emptied. Which was made the default behavior in order to facilitate automation in vanilla. I propose that something similar be the case here. Instead of water being pulled from what is essentially the input slot, the output fluid should be the only one that can be emptied. Were that the case, the set up above would work just fine. Water would flow into the brewery and the output fluid from the brewery would flow into successive processing blocks until finally entering the bottler at the bottom left.