I'm reasonably sure USD (or EUR) will still exist, have some reasonably similar value, and be usable in five years.
Sure, but even with an inflation rate of 2%, if you hold USD for 10 years, its purchasing power is cut in half.
Every BAT token that’s ever going to exist has already been minted, so it can’t be inflated by creating more of it. There will be no quantitive easing, “too big to fail” or bailouts in the BAT ecosystem.
That almost certainly will not be the case for either the dollar or the Euro.