When you say the radio started to work badly, in what ways? Sometimes re flashing data will clear these things up. Possibly there is a software update from Mazda for your vehicle. Could be something as simple as a loose or failing plug.
The Car's OS is tied into the radio in my CX-5. I suspect similar in the CX-3. For that reason alone I personally would not tamper with installation of an aftermarket radio, especially a no name one.
Some have cracked the Mazda radio code and were able to add things like other apps and skins to their existing radios.These were older models and not 2019, so things could have changed making this more difficult.
I went through a phase where I was into car audio. I mean REALLY into car audio, I installed systems on the side and put new systems in all of my cars, even worked for a company who did it as well for awhile. Back during those times there wasn't as much trouble you could get into. Sure if you didn't understand the wiring you could smoke your radio or even cause a car fire, but you had to be pretty dumb to do that. I always say, if you don't understand it, then stay away from it, unless you care to learn first. Don't learn the hard way which might end up costing more $$.
Most car audio now uses factory plugs making it far more difficult to have those kinds of problems HOWEVER- much much more of the car's OS and operational functions are often tied through or coming from the 'radio' of the car. If you remove a radio you might not ever get your car started again unless you take it to the Mazda dealer.
If you want to be the guinea pig here though, I would be interested to see how it works out. If you do it, please let us know how it went.