The match mod in the advanced filter doesn't work with Exotic Birds. I don't know if it's caused by something you can fix, or if it's an issue with Exotic Birds.
The filter is set to Input & Output, Allow, Match Mod, Ignore Durability, Ignore NBT, using Egg Analyzer as the filtered item.
It doesn't even allow the Egg Analyzer through the filter.

I did a bit more testing and it seems like the storage controller was putting the items in a different storage that was further away for some reason. Removing the unfiltered storage allowed the filter to work.