All I can say is there are a lot of various complaints with ReLive. Feel free to talk to AMD support about the issue:
https://www.amd.com/en/support/contact-email-form
Maybe another user can help if my advice does not, so apologies up front.
If you have not done a clean install of the driver you might want to reinstall the driver with that option.
Next you can install the driver without ReLive installed. Many users report that not installing ReLive alleviates other issues they have.
Next there are other software packages that are free that most users find work much better than ReLive. You might wish to explore that if you have not.
I do agree with you, everything about your purchase should work at default settings out of the box and yes it is very commonly reported that, this does not happen with AMD GPU products for far too many reasons. I hope that eventually AMD fixes this as the end users deserve what was promised.
OBS Studio is an excellent free streaming software but it comes with a IMHO non intuitive interface and does have a bit of a learning curve. That being said, it is in my opinion better that most the paid software.
Good news is that some other folks make a user interface replacement for OBS Studio and it make using it much easier. That software is called, Streamlabs OBS.
Good Luck!