Hello Lawrence, thanks for the suggestions.
Yes, I did all the above. The manuals for these devices can be found at:
https://usermanual.wiki/Homewerks-Worldwide
Mine is the 7130-06-BT, which the instructions for pairing can be found at page 10 of its manual:
https://usermanual.wiki/Homewerks-Worldwide/7130-06-BT/pdf
The instructions are quite poor and basically only mention the steps related to your BT device (iPhone in my case) but says nothing about the steps regarding the actual Homewerks device. I tried the manuals for other Homewerks devices but the text is basically the same.
Fortunately, I was able to find other references in forums and was able to pair all iPhones we used to have in the house. I paired one iPhone 7, two iPhones 8 and one iPhone 10, but the same pairing procedure did not work for the new iPhone 12.
Basically, you turn the switch on and press the "music" button. You will hear a ringtone and it will connect automatically with the last phone it paired with. If you have a new phone then you will have to turn the speaker off, then on again and do a long press in the music button until you hear the ringtone. It also worked with two short presses (I found this empirically). Once it pairs, then it is just a matter of pressing the music button once to have it paired with the last device.
When trying to connect with the new iPhone I did try various combinations of pressing the button, hearing the tone, turning on and off BT in the iPhone, etc. Nothing worked. All works for the iPhone 8 I still have.