Can someone with some legal knowledge explain what the process is for a legal precedent to be undone? What does a 'petition' to rehear a case actually mean? Is that a thing that might feasibly happen?
Petitions for rehearing are not granted routinely, but reasonably frequently. Probably better odds than seeking Supreme Court review if you’ve got a case where there is no circuit split (disagreement among different courts of appeals). The judges in the court all receive the rehearing petition, and they can vote to rehear the case. If a majority think the case is important and the panel got it wrong, they’ll grant rehearing, in which case the case will be argued again in front of the whole court.
One important thing is that a panel can rehear it’s own decision, but can not overturn the ruling of a previous panel in another case. An en banc court can overrule any of the court’s prior precedents.
[1] Except the ninth circuit, which is so big and geographically dispersed an en banc Court is something less than the whole court.
If a situation that all lower courts are getting "right", then there is no need for them to think about it, the lower courts are all getting the same answer, and the lower courts are all made up of smart people who have thought about it. If there is a situation where the lower courts do not agree - that means smart people have thought about the issue and do not agree, thus they need to think very hard about this situation to come up with a perfect answer. Perfection is hard of course, so they generally narrow the scope of what their decisions cover until only the minimum possible scope is covered thus ensuring there is less room for a subtle situation where they are wrong.
TL;DR: Precedent is binding to court where it was set and "below". To overrule it, you generally must appeal until you reach the level above where it was set. IANAL.