Why do Bluetooth modules always include the whole protocol stack
For certification, the hardware and the software must adhere to the Bluetooth specification.

The whole point of modules is to make integration of Bluetooth as easy as possible. So the module gets certified according to Bluetooth specifications and local radio specification (FCC and all that stuff all around the world). With this, you can integrate the module very easily into your product and save a lot of costs (>50k for certifications).

But to get the Bluetooth certificate, the Bluetooth stack has to be integrated into the module already as it is part of the tests (there are strict timing specifications for example).

So if you want to write your own stack you end up with a custom solution which you have to get certified. In that case you'd rather buy a chip and design your own module, so you can save at least some money on the hardware side in the long run.