A race isn’t interesting if isn’t competitive. But beyond that, the centrist impulse is definitely to bolster the weaker side, especially if that side is unattractive and feckless.
I haven't come across a term that covers what we're talking about in full scope(?)(hence my clumsy tug of war metaphor), but that's probably because I am peripheral to game theory / not in the psychology space
(We have lots of terms for the emergent effects - Overton window, Political Polarization etc. - But possibly not for the overall phenomena of why polarization occurs?)
They are very worried about being fair-minded. They see a vocal minority as social proof that the minority viewpoint is sincere, thoughtful and worthy of consideration. Neither is a bad assumption to start with, but it's not a great place to end the thought process.