Sending data using Bluetooth SPP-C Module without a Breakout-Board
I found a solution myself.
The modules are working as they are supposed to and are not faulty.
Also I did nothing wrong.

The Problem are my development machines. Mac mini (Late 2009) and MacBookPro (Mid 2009), both having a Broadcom 2046 Bluetooth host chip. I used those to connect via Bluetooth and open a tty session using minicom. Running macOS and using "Hardware IO Tools" to log Bluetooth packets I saw that there was clearly nothing coming in.

After that I turned over to my Windows and used Hype! Terminal. Everything worked!

Then I went to my Raspberry Pi and Linux. After pairing using bluetoothctl and issuing a few commands from Bluetooth as ttyUSB. Everything worked as well.

Then I tried BlueSPP on an Andriod phone. And again, it worked as it was supposed to.

So my conclusion is that there is some sort of incompatibility between the SPP-C module and the Broadcom 2046 host controller.