Even in the past, I never played a Mega Drive game with that issue. I thought Retro-Bit added those buttons which I presumed for modern gaming reasons.
For my Pro Controller, I had the "Select" button assigned for Mode.
So, if I ever emulate a Mega Drive game and find the controls are screwy, just press mode to fix it?
I don't know for emulation, I never have problem when playing it emulated.
May be it's not an issue anymore.
But in my real console, I did encounter that problem when playing Golden Axe II.

The solution back then it's not just pressing MODE button, but holding down the MODE button on the controller while turning the console on, and releasing it only after the SEGA logo appears.