The Hardware Modification

First of all you have to open the back cover of the remote control and as next step the additional cover with the nameplate. Now you have the back side of the PCB in front of you with all the test-points. You can now also remove the PCB, but that's not needed! All you need are the test-points which directly connect to the buttons. The following picture shows you the relevant test points:

The buttons are connected via pull-up to the supply (around 3V) and are pushed to GND by the buttons.

Now I added open collector circuits in parallel to control the signals like the buttons. Thanks to the open collector circuit we have a logic OR condition between the hardware button and the input signal. Additional the open collector circuit acts as a level-shifter, so a 2.5V .. 5V microcontroller can be connected (e.g. the 5V Arduino Nano).
The collector of the NPN transistors (e.g. BC846) must be connected to the test points, the emitter to GND (e.g. by scratching of the solder resist and using the GND-plane or by using a wire to the GND battery connection). Don't forget the resistor between the input and the base.Hint: In the internet there are various projects which directly connect the test points to a microcontroller (e.g. ESP8266) GPIO port. I wouldn't recommend this solution as you now will short the GPIO port to GND if you push the button (exception: You can configure the GPIO port to open-collector like some STM32 can do). If you nevertheless want to use this solution: Check that your GPIOs are working with 3.3V.

My compact implementation:

Don't forget to securing the cables to relieve the solder joints.

I had to modify both covers for the cables:

Software part
As next step we must perpare the Arduino to understand the Firmata protocol (via UART).
At first install the Firmata library:

Now open the StandardFirmata example:
File -> Examples -> Firmata -> StanardFirmata

Select the right COM port and model and flash the file to you Arduino. Now your device is ready for communicating via Firmata.

Now we must prepare the other side, the Homa Assistant system. First if you use Homa Assistant in a VM check that the USB device with the virtual com port to the Arduino is passed through to your VM. In OMV it looks like this:
Now check if the device is available in Home Assistant by going to the settings -> system -> hardware -> all hardware (Hint: Advanced Mode must be enabled in your profile).
Here you see the name of your device:
Now you can activate the Firmata integration in HA by editing your configuration.yaml like described here:
https://www.home-assistant.io/integrations/firmata/
You can use the "File editor" for editing. You find it in the Add-on store.

Change the name of your serial port device (see the entry in the all hardware list) and configure the ports as you wish. I connected the first roller shutter to pin 2, 3 and 4.
If everything goes right, you will see the Firmata integration your integration overview:

You can now control the ports:
To make buttons out of your switches you can use input_button helpers.

These input helpers now trigger automations that turn on the outputs and turn off them again after some hundered ms (like you would push the buttons on the roller shutter remote control). That's how one of this automation looks in YAML (but you can also configure it in the GUI):

Add the input_button helpers to your dashboard and you're finish: