But, again, shifting in the mainstream literature (and thus, in the way Rothbard uses the term), means “shifting the reduction in income from a tax levied on you to someone else.” The sales tax is levied on the entrepreneurs. The state collects the tax from them. But the entrepreneurs’s income is profit, the difference between his selling prices to consumers and his buying prices to owners of factors of production. This leaves open the possibility that entrepreneurs can restore the pre-tax price spread (and thus, their profits) by either raising their selling prices for output or lowering their buying prices for inputs. If entrepreneurs shift the tax, then their income, which is profit, remains the same even though they pay the tax while that of either consumers or owners of factors of production declines.
Shifting does not refer to the production dynamics of the market. So Rothbard’s claim about the production effects of the tax are subsequent to new pattern of prices of factors of production result from the tax. In response to your last statement, then, Rothbard would say: the reduced demand for factors of production is the method entrepreneurs use to shift the tax to producers. The resulting change in the pattern of prices for producer goods will make some previously-viable production now not viable. With reduced supply, given demand, the price for such goods will rise. The rise in price harms consumers but does not shift consumer incomes to the state.
The attempt to shift the sales tax forward to consumers is asymmetric to the case of shifting it backward to producers. In the forward shifting case, when the tax is levied on entrepreneurs, they have previously-produced stocks to sell. It only reduces their revenue to raise their prices. Therefore, they will not do so. Their revenues are maximized at the existing prices because when the state levies a tax on entrepreneurs it does not increase consumer demands for their products. According to Rothbard, then, taxes can never be shifted forward. So there is no such case to compare to the case where taxes are shifted backward to analyze whether or not firms go out of business in both cases.