Low intensity colors are a bug in the actual cartridge of the game as documented. There is a patch in romhacking.net (https://www.romhacking.net/hacks/4792/ to fix it with this description about the bug:
Back to the Future: Part III, for Sega Genesis / Mega Drive, is a game that is
broken in a particularly weird way. Genesis games use a 9-bit palette, with 3
bits each for Red, Green, and Blue color intensity. These are stored in a 16
bit "word" (2 bytes) using the format:
Somewhere in the development process for this title, the programmers got things
mixed up, and mistakenly stored all palette data - and code to modify colors -
in the wrong format instead:
In other words, all bits shifted right by one place. The result is that all
colors on screen are reduced to *half their intended intensity*, making the
game look dim, and with low contrast. How QA missed this is beyond me... I
guess their TVs were just turned up really, really bright.
In fact, there is one place in the game that DOES have the correct palette:
the SEGA logo that displays when the game is first turned on. It's likely the
logo, and the associated "color fade" code, was taken from SEGA documentation.