No. Almost Yes. Yes, flat tax seems fair; but No, because it burdens the low income taxpayer.
Add one component to the flat tax system to make it fair: an offset. Put on your math hat while I explain a "linear tax system".
The linear tax system has 2 components: a "slope", which would be a flat rate; and an "offset", which would the adjustment taxable income before applying the slope.
The tax rate (slope) would be applied to taxable income. Taxable income is the total income less the adjustment (offset). The adjustment would be based on a basic allowance for the taxpayer and dependents, and other taxable income credits.
To contrast the linear system with the current progressive tax system, the linear system , similar to a flat tax system, has a single rate. To contrast the linear system with a flat tax system, the adjustment provides relief for low income taxpayers, while providing incentive for everyone to spend - which means economic growth.
Adjustments should be allowed that benefit our economy and society, and should be allowed equally, across the board, with no restrictions or limitations, but never to reduce taxable income to less than zero. Similar to the current tax system, adjustments would begin with what is currently referred to as the "standard deduction", and with deductions for dependents as well.
However, adjustments would be broadened to provide incentive to stimulate economic growth and contribute to the betterment of society and our world. The decision of what adjustments would be allowed should be made by an competent, impartial (i.E. Neither Republican nor Democrat; perhaps a group of centralists) legislative branch.
A flat tax system would only burden the low income taxpayer, only to contribute an insignificant amount of tax revenue. The progressive tax system promotes stagnation in its unfair increases in tax rate as income grows.