Media keys behaviour with iTunes in macOS Mojave
Solution 1:
I went through almost exactly what you went through. I fully reinstall Mojave with no luck, so just downloaded the Media Key Forwarder (note: it's no longer called Media Key Enabler). I recommend it until Apple fixes their 'unified play'/'Now playing' mess.
Update
Boot into safe mode (hold shift as you press the power button, or click here for instructions). Start iTunes and the play buttons should work, then reboot normally and enjoy life.