The key is not to fix the entire budget crisis, its to fix a single huge problem that by itself consumes the majority of most state budgets and usually more than that. It is a huge problem, there is no denying it.
Public schools spend more than private schools and are grossly inefficient and mismanaged.
A major example:
They don't account for actual costs, and when they do the figure is underestimated by 44% on average.
That
cato report (specifically on pages 3-14) I linked you to earlier details how accurate and up to date financial reports are few and far between and that in some cases public schools were spending 90% more than what was stated. The information was obtained from the NCES, and district and state budget reports.
So yes there are fixed costs however the actual cost is often misrepresented and the result of extreme resource mismanagement and also does not take into account what changes will result from competition.
And to play on your example of a wooden shack with a candle on it, why couldn't a certified teacher decide to turn their home into a small school and use the voucher money to purchase the licenses, materials, and equipment he/she needs and still turn a profit and meet the educational standards with a smaller size class? Currently that is nearly impossible to accomplish.