The other answers address some of the reasons why, in this case, inertia of the status quo is not overcome. From the perspective of the IRS simply mailing a bill creates a collection challenge that is more logistical and not purely political.
Most income Americans generally make through wages is reported (and witheld) directly to the IRS as it is being made, and most businesses are required to make earnings and distributions reports available and payments made to the government quarterly. However, not all income is reported quarterly since the underpayment penalties for some circumstances is small, and America is somewhat unique in that it currently also taxes income from worldwide sources. This means that there could be significant tax liability that exists which the U.S. Government has no way of regulating directly. This is one of the arguments against citizenship-based (versus residency-based) tax laws.
Another complication is deductible expenses which may disappear in a simplified tax plan, but probably not completely. The government will not be able to determine prior to you declaring your intent to deduct any given expenditure, so it will not be able to include those factors into a simple "bill" to be mailed and then paid directly by each resident/citizen. As Adrian McCarthy correctly points out in the comments, the amount of information the Government would need to collect and also make decisions about is vast. Farmers for instance can deduct livestock's depreciation over 5 years as a business expense (source), but have the option of front-loading all of that depreciation in the first year after purchase. This makes sense if you're planning on a bumper crop and you want to hold on to those particular animals for five years, but you may want to decide not to do that if you think your income may be higher two to four years from now, or if you want to sell an animal before 5 years have passed and not have to track the basis and repay back in the year it is sold. Sometimes, two animals meet in a field and a new animal is produced, but this animal has different rules for you because it just so happened to be born on your farm.
Goods are sometimes bought for cash at auction where aggregate information may be provided after sale but individual buyers are not identifiable, and in order to calculate everyone's deductions correctly the government would have to know this information. Other questions that may affect deductions such as "Did you buy a house this year?" are easy enough perhaps for the federal government to track, but each one is generally complicated by a related question such as "Is this your primary residence?". This creates a decision making process that is intractable from anyone's perspective other than the individual tax payer, or would require an amount and sophistication of tracking and reporting that I don't think most people would be comfortable with.
You could get around this by just mailing the bill and then have a place where you could list any deductions and calculate a different bill amount, but then you'd have just about re-invented tax filing. It can be much simpler, but politicians have yet to show the will required to make it so.
Politicians ranging from Grover Norquist to Ronald Reagan have also been champions of the idea that, in Reagan's words, "paying taxes should hurt." From that perspective, letting the IRS "calculate" your taxes so they can then send you the bill is a conflict of interest since the Government will of course overcharge you. Instead individuals should be given the chance to calculate the total ourselves.
NB- I don't think being able to make certain deductions from your taxes will ever be able to disappear completely because politicians are pragmatic people and they know that upsetting everyone will probably be the last professional politicking they'll ever do. Upsetting other people's constituencies though is something that can be tolerated, and there could be actual practical uses for using the tax plan to encourage certain spending habits from time to time. Plus the sheer weight of it all has created a few businesses that are optimized for it, and could cause a systemic shock for them and their customers/creditors if it is ripped away like a band-aid. All of this creates uncertainty, which some could argue is also the government's duty to prevent.