I'm comfortable (but not upper) middle class. Moving on.
The dream is dead and beyond dead. It never existed. Nice idea? Hell yes. Ever true? Not so you'd notice.
Yes, if one works hard, is very smart, and a little lucky, you can climb.
However, the "luck" part is the problem. In order to seize opportunies, they have to be there. And generally, they aren't. Particularly as more and more jobs that actually pay decently demand college education.
So Joe Shmoe, who puts in a 14 hour day, may be able to pass on something to his children, but often, its a tale of how despite one's best efforts, they aren't enough.
As for fatcat CEOs? Using them as an example of the American dream being true for more than a handful at most (and most of them come from families already in a comfortable position) is absurd. If the American Dream was true, then I could take Jose off the streets, give him a passable education and good wishes, and he'd be able to work his way up to any position he wanted.
Odds of that happening? If it ever was possible, it was in the 19th century, when there was a lot more "uncontrolled" for the fortunate to grab and make their own.
If John has 80% of the wealth in a given region, then John should be paying 80% of the taxes collected in that region. With all the various ways the wealthy can avoid being taxed fully, he almost certainly won't.
As for reducing the scope of the government: How far can you reduce it before it is too limited to do any good? The bloated beaucracy concept has less foundation in reality than the flat earth theory and creationism.
As for "fair share"...when and if the rich only are rich because of their own efforts, not luck, not the system being unbalanced, not any other advantages they don't deserve...we can talk about letting them being rewarded. So long as you're embracing capitalism, forget the rich earning their wealth purely by their own sweat (metaphorically or literally). They earn it by exploiting...people and/or situations and a healthy chunk of good luck (which I don't begrudge them per se, but is not something they deserve any more than I do or the poor do).