That said, my sense is that most people don't know what their total yearly tax is, nor their tax rate is, only their refund/tax payment that's due in April 15. As the Republicans found out recently, a lot of people complain about their "tax increase" when they don't don't get a refund, even if their actual tax rate went down.
* Special interest groups who have won tax credits/deductions--most of these don't get reported to the IRS, so it makes it easier to kill these off since fewer people will take advantage of them.
* The anti-government crowd, who is a) afraid that the IRS could abuse this power to tax people more (if people get used to signing the dotted line, they might not notice if the IRS stops accounting correctly for some reason); b) adamant that people know just how much the government is "stealing" from them in taxes and so form a ready constituent for lower taxes; and c) generally don't like the idea of the government telling them what to do anyways, so the government just sending them a bill is the ultimate anathema.
* People who are underreporting certain classes of incomes who might be afraid that there will be a push for broader mandatory reporting to help make taxes even simpler.
Given that the political power of the second class I've mentioned is quite vast, they are sufficient to prevent the sensible implementation of the IRS-sends-you-a-bill even if TurboTax and the like were to stop lobbying the government.