The flat tax doesn't "in theory eliminate deductions and loopholes" any more than a progressive tax does.
It's just one particular scheme for setting the income tax rates. One can eliminate loopholes - or not - independently of that.
And no politically feasible scheme - flat or otherwise - is going to do away with said loopholes, any time soon. And the advocates of a flat tax know this. That's why it's disingenuous of them to conflate the two issues: they know their suggestion is basically a give-away to the rich that harm everyone else, and so attempt to fool the rest of us into supporting it by conflating it with simplification of the tax code, which isn't on the table at all. If what they cared about was closing loopholes, they'd be suggesting that directly. But they aren't, because that would take money out of their pockets.
After all, if you aren't rich you must be stupid, right?