Even though the radio is on when I turn off the car, it’s often not on when I come back and turn the car on?
Check your owners manual. On my van, the power to accessories stays on for something like 30 seconds after the ignition is turned off. During that time one could close windows and sunroof, etc. You could sit in your car after turning it off and see if the radio goes off after a time. Opening the door may also be a factor.
Radio not on when starting car, that is a good thing if there is a teenager in the household . That way your eardrums won’t be shattered by the excessive volume of the radio.
Have you had any trouble suggesting a failing battery? I suppose if it’s starting you are not having that problem, but one doesn’t know if one doesn’t ask.
The owner’s manual on the 1954 Buick I once owned stated that the radio should be turned off before starting the engine. In that car, the ignition switch had 3 positions, lock,on, and off. The starter was activated by turning the ignition switch to ON and stepping down on the accelerator. The radio was not connected through the ignition switch and could be turned on whether the car was running or not. Operating the starter could damage the vibrator “or pulsator” which allowed the 12 volt DC battery voltage to be stepped up to supply the higher voltage to the plates of the vacuum tubes. Now I know on later model cars have the starter switch built into the ignition switch and that the radio and other accessories are not powered when the ignition switch is turned to start, but I still think it is a good practice to turn the audio system off before turning off the ignition.
Visit a dealership and ask if you can test how that function works on one of their cars. It may just be designed to do that. My guess is that it is a minor software glitch, a software programmer didn’t notice it at the time the software was released. So check if they have any software updates available for the audio system software while you are there.