You said "... why do people believe the rich don't pay their fair share when the top 10% pay almost 3/4 of the taxes... "
I'm suggesting talking about the top 10% is misleading because when people talk about the rich not paying their fair share, they're not talking about the master electrician in your home town that broke $180k working overtime. You conflated the rich with the top 10%, but your grouping includes a whole lot of people that most of us wouldn't consider rich.
To answer your question about why people believe the rich don't pay their fair share, you either need to look at a different figure than the top 10% (which invalidates the 3/4 of the taxes portion) or you need to say everyone (including married couples filing jointly) making $178k MAGI is rich. I don't think people at large agree with the latter, so the rest of your question is premised on the wrong set of numbers. That may answer the question you posed initially.