I am sure this will be an issue with a lot of super partisans. My neighbor still believes that Obama was not eligible for being president. I have no idea what evidence could convince these people otherwise.
I think the people who are spreading these rumors are playing a very dangerous game. There seems to be a trend to spread doubts about elections, FBI investigations, CIA findings and others without providing real evidence. This is a short term benefit for the people spreading the rumors but I wonder if they are starting something that may spin out of control at some point.
You talk like you have been living under a rock for the past three years. Trump got politically famous through the birther movement. Most of the people who support Trump believe in a lot of these rumors, as you call them, or fake news, as the rest of the world now call them.
Fake news has been a big problem around the world for the past few years and the media has acknowledged and widely publicised it as a problem since 2016. Since a lot of people get their news from Facebook now, and Facebook has been algorithmically recommending fake news articles to people, Facebook has gotten a lot of flak for it, including the CEO being called to testify about it in Congress. A lot of people are now trying to figure out how to combat fake news and this has been on the news and this website for the last few years.