But you were not able to hold a connection between the computer and the HC05. I did not have the same experience, and before we go through things step by step, we need to rule out the possibility that you have a defective HC05.
The best way to do that is to confirm the HC05 stays connected with an Android phone. If you don't have a friend with an Android phone you can probably get a $20 pay as you go Android burner phone. You will not need any phone service, as the bluetooth part of the phone is independent from anything that uses messages, data, or voice.
The fact that you could make a short connection indicates to me that the PC set up was correct to pair the device and arrange the port. Certifying the good health of the HC05 is really a necessary step,
You will not need the Android phone for any longer than it takes to pair with the HC05 and confirm that you can connect and stay connecyed with Morich's Serial Bluetooth Terminal App. Then you are done with the Android phone and it back to trying to get the computer to stay connected.
I have no experience. If you can get anything to stay connected to the HC05 and confirm that the pc disconnections were not due to the HC05 module being defective will be of value.
As I have said, following the procedures you were also following, I could pair, connect and communicate between Win 11 Bluetooth and the HC05.