Jesus.
A flat rate is not the same as a flat tax.
A truly flat tax does not have exemptions. It starts from the first pound and keeps on going to the last pound. As Wikipedia puts it (admittedly not Tolleys, but they do have a lovely turn of phrase) "any flat tax with an initial threshold deduction is inherently progressive."
Perhaps you (and the OP) meant something other than a flat tax - ie a tax with an initial exemption and then a single rate thereafter. But that's banding. That exemption has to be set somewhere. If you set it too low, not only are the poor (who, desperately unfairly, still get a vote each even though they don't pay in as much) unhappy - but they can't afford to live. Poverty (which costs public money to remedy) goes up, the workforce becomes less internationally competitive, and bottom end tax evasion and avoidance increase (black markets, cash in hand, squatting etc). But if you set it too high, you give people an easy ride, then you miss out on valuable tax revenue - remember, though one per cent of earners are paying the 50% rate, nearly 100% are paying 20% and that money adds up.
So what's it to be? Most governments choose to tax on stepped bands so that it hurts everyone, but no one gets hurt more than they can take.
You don't need to be an economist or to read Tolleys to understand this. You just need to watch "Scum". Ray Winstone's character has a conversation with a sidekick about the taxation of other inmates' income. They are operating a genuine flat tax - 50p in the pound. They realise that if they set it too high people won't pay and, implicitly, will allow another "daddy" to take Winstone's place.
One of the many things I didn't learn at Oxford ...