I'm on the fence about that one. On the flip-side:
1. If they don't police it, then Facebook as a whole gets a reputation for disinformation. (When I look at Facebook, I personally ignore most shared articles because they're mostly crap, and I don't want to look at them. I only care about the personal goings-on of friends, and the signal-to-noise gets worse over time.)
2. If you take someone who doesn't care about truth, they're not going to introspectively seek-out a metric which might report "You're misinformed and stupid!" BUT If you're self-righteous about the truth, then you might want some objective metrics which say, "Look I don't spread misinformation or live in an echo chamber! But Cheryl does! Look, Cheryl, I'm smarter than you!" That might provide some basis for a viral dynamic.
Remember on your first point though that it might not be a factor for the majority of people. Can also reach people through ad networks, or social networks, etc.
Facebook could probably give people a public rating based on the accuracy of the information they shared, and shame the worst offenders, but neither they nor their users would have an interest in this being established.
I don't like the idea that you can play a straight bat but be overrun by hustlers.