The only problem I see this causing is that the potion satchel can basically act as stacked potions (which are super OP). Have an inventory full of instant damage and have a single satchel slot selected and you can rapid fire those things. Currently a player must sacrifice most of their hot bar if they want to potion spam, which lessens the severeness of the problem of potion spam. With an item like this, a player could become so deadly that they would become unstoppable.
I don't do PvP in Minecraft and so I only know what I read about it on this board.
After reading your response it seems to me like there's a problem with potions itself and my suggestion just amplifies the severeness of the problem.
If I understand correctly you can put 9 damage potions on your hotbar and - depending on skill and latency - you can fire those in rapid succession to deal horrible damage to you enemies.
So if you fill your whole inventory with potions you could do this 5 times and after that you'd be completely useless in a PvP fight without any means to replenish your stock.
Sounds like a pretty unfun way to play and is only a viable option because it's so OP.
If I were a developer working for Mojang and wanted to overhaul the combat aspect of my game, then this is something I'd change first.
Maybe add some sort of short cooldown to potions (per effect), so you wouldn't be able to spam them like crazy.
I know, this isn't very creative, but I'm sure Mojang will have some better ideas for it.
Balancing potions might even allow them to stack, which I personally would like to see happen.
In case Mojang really neglects balancing out potions then I'd add a 5 second timer per item slot on the equipped hotbar, when a potion is used.