"So why has this proposal been largely absent from the public debate? I have no clue. "

From The Economics Of Contempt, we have another sales tax decrease supporter:

"Suspend State Sales Taxes!

The main point of a fiscal stimulus package is to offset the decline in consumer spending, which accounts for over 70% of GDP. We need to avoid the Paradox of Thrift, and all that. As Paul Krugman put it:

[W]hat the economy needs now is something to take the place of retrenching consumers. That means a major fiscal stimulus.

The argument against tax cuts as fiscal stimulus is that consumers won't spend the extra money, but rather will increase savings or pay down debt. We want people to spend, and spend now.

So why oh why aren't people talking about temporarily suspending state sales taxes as a way to quickly increase consumer spending? Suspending state sales taxes would encourage consumers to spend more by lowering the price of spending, and making the suspension of sales taxes temporary would encourage consumers to spend before the sales tax holiday is over—that is, to spend right now.( CORRECT )

Laurence Kotlikoff and Ed Leamer have offered a very nice sketch of how such a policy would work. The proposal has been endorsed by Stanford's Robert Hall and Susan Woodward, and was evidently quite a hit at the annual American Economics Association meetings, winning converts like Alan Blinder of Princeton. Here's Kotlikoff and Leamer's overview of their proposal:

A better way to spur consumer spending is for Uncle Sam to run a six-month national sale by having a) state governments suspend their sales taxes and b) the federal government make up the lost state sales revenues. The national sale could be implemented immediately.

Here’s how it would work. Uncle Sam would pay each state a fixed percentage — say 5 per cent — of the 2007 consumption of its residents. States would be required to reduce their retail sales tax rates by enough to generate a six-month revenue loss (calculated using 2007 data) equal to the amount they’ll receive from Uncle Sam.

For states with low or zero sales tax rates, implementing this policy requires making their sales tax rates negative, ie subsidising purchases. Shoppers would see a negative tax on their sales receipts, lowering their outlays. State governments would reimburse businesses for paying the subsidy and, in turn, be reimbursed by the Feds.

States would be free to broaden their sales tax bases to apply the National Sale to all retail sales, not just the sales currently covered in their sales tax systems. To make the policy progressive, states could also reduce sales tax rates by more for goods and services that are disproportionately consumed by the poor.
...
[This plan] will apply economic medicine where it’s most needed – on consumer spending, giving everyone an incentive to spend now and begin again to trust our economy and its institutions.

So why has this proposal been largely absent from the public debate? I have no clue. "