I never mentioned these people.
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.
Why?
Because that enormously expensive army we're maintaining is protecting mostly their stuff. Setting aside the fact that, say, poor people, have no money; why should the poor pay an equivalent share to protect the wealthy person's property?
I'm not really sure what you're trying to argue for/against here. Yes, people in the top 10% of income-earners collectively pay the majority of income taxes, dollar for dollar, and yes, we've been hearing variants of what Jonathan Chait called "The Stat" for years: the highest-earning 1% of taxpayers pay 40% of all income taxes.[1] As Chait points out, "'The Stat' is literally true, but it is deeply misleading." For instance, FICA is not a progressive tax; it's a flat tax that stops being collected at around at around $150K of income. Somebody making $80K pays way, way, way more FICA as a percentage of their income as somebody making $800K does.
While I'm not suggesting we need a wealth tax, start burning down mansions, etc., etc., it's at least worth considering the possibility that America has a tax system which disproportionately favors the wealthy. That multimillionaires and billionaires pay, dollar for dollar, more taxes than the shift manager of your local Jersey Mike's does is not some kind of slam-dunk argument against said possibility.
[1]: https://nymag.com/intelligencer/article/fact-check-richest-1...
I hope you're just confused about what everyone is saying, and not being intentionally disingenuous.
> if the link had a 0.1% bracket it would show that the even smaller group pays an even higher percentage of the total income tax revenue relative to their size.
That's exactly the point: in absolute values, if you have a lot of concentration of wealth, those will come out to be most of the tax base. That doesn't mean that proportionally they are paying the same. As a matter of fact, someone that makes 100k being taxed 20% is actually more onerous than someone that makes 1M being taxed 20%. This is why most countries try to have a progressive tax system.