Why would an empire living in the medieval era with medieval Europe era technology choose to use(and continue using) wooden cannons in spite of advancements in technology and metal working?
The only strategic advantage I can think of would be mobility. A wooden cannon (including normal ammo and metallic holdings) would be lighter to move about than it's metallic cousin. I also believe they'd also use a mixture of wooden and metal (bronze, cast iron or steel) cannons depending on the situation, the Quaker Gun would be one of these application, allowing for a regiment to seemingly bolster the strength of their unit for a cheaper price.
Yes, there's a Mythbusters episode about whether it's possible to make a cannon out of wood. They used iron bands around the barrel to ensure it didn't go Boom! till they wanted it to (and it being Mythbusters, of course it went boom).
They were able to put multiple shots through the cannon and could have continued to do it indefinitely.
It's already been done almost exactly as you described.