I have 5 hoppers and 5 double chests all ontop of eachother, the hoppers are all connected to the chests stacked ontop of eachother, the items from my farm flow down the hoppers into the very bottom chest but when that chest becomes full it fills the hopper and then drops the items and won’t put them into the next chest.
You need to place the hoppers so that every one of them points into a chest. Items will fall to the bottom, filling the bottom chest, until the bottom hopper backs up. Then the next to bottom chest.
It sounds like you know this already, maybe double check that the hoppers don't point downwards. Even then, stuff should never fall out the bottom until the hopper jams up, then things can spill from the top.
An alternative stack is to zig zag the chests so that you can place hoppers under each one that feed into the next chest below. This makes items go through every chest till they get to a chest whose outflow is blocked - makes it nice that the bottom chest keeps refilling from the ones above.
I have to say though, the stack you show should really work.