It's not bias to call that a baldfaced lie.
Barack Obama proved he was a citizen to the extent that any of us do: he had official documentation which was adequate to prove citizenship to the government (for example, producing a birth certificate as a part of the documentation package for getting a passport). He publicly released it. Then the goal posts moved to talking about long form birth certificates (I personally had no idea that there were different birth certificate formats prior to that). He also released that. And of course nitpicking all of the documents when they were released. Obama bent over backwards to humor a bunch of people propagating a common racist smear (questioning the citizenship of any American who isn't white).
And all of that aside, it doesn't even make sense as a conspiracy. The Democratic establishment wanted Hillary Clinton to win the primaries and had their thumb on the scale for her (by the time 2016 rolled around, they'd crushed almost all of the competition that might've considered a run and rigged things against Bernie so another Obama upset couldn't happen). To what end are they running some grand conspiracy to back a candidate they didn't even want?
Trump's claims of voter fraud went to courts across the country, failed to produce any evidence, and got tossed. They even got debunked by his own appointees, like noted communist Bill Barr. The media entertained the discussion far longer than they should have, because Trump & getting people wound up is great for ratings and article clicks. It's not a particularly good example of bias, but it is a really good example of the toxic nature of media (with media outlets intentionally stoking outrage among their target audiences).
Anyway, fuck Rush. He was trash. He's a big reason our current media landscape is such trash.