We have a "voluntary" tax system in the U.S. -- that's always been the situation. "Voluntary" doesn't mean that that you can choose to not volunteer to pay your taxes. It mostly just means that the way we run things, by default, it is each citizen's responsibility to calculate and pay their taxes each April.

American taxpayers filled out 1040 forms in the days before computers, a lot like they do now. The IRS selected certain fillings for audits, just like they do now -- sometimes because of an apparent discrepancy, and sometimes just at random.

It would be a lot more work, take a lot more resources, and be prone to a lot more error and lawsuits, if the IRS tried to calculate everyone's taxes for them. Even now that we are in the days of computers, it is much more efficient for the IRS to only audit a fraction of the filings submitted each year.

I'm also pretty sure our "voluntary" tax filling system has something to do with the Fourth Amendment and other privacy concerns. A lot of Americans very strongly believe that it is not the government's place to be all up in their private business.

-- EDIT to add:

There is a difference between whether it would be possible for the IRS to calculate individual citizens' taxes and whether we should abandon our voluntary tax system for one in which the IRS simply calculates the taxes owed by every citizen and send us each a bill. My original response was intended to address the latter, but now I'll say something about the former:

For someone whose single source of income is a job working for someone else, of course it is possible for the IRS to calculate your taxes. You've already volunteered all the information the IRS needs to do so. Your employer has already told the IRS exactly how much income you've earned and exactly how much of it you've had withheld for taxes. Remember when you signed that withholding paperwork with the HR department on your first day? That was the moment when you personally volunteered your income information and payments to the IRS. You've literally already been reporting your income and paying taxes on it ever since.

The way taxes work in practice for a single-income employee does not reveal the potential complexity of tax accounting for individuals who are self-employed, who have multiple sources of income, and anyone who doesn't want to make regular fillings and withholding payments throughout the year. The tax situation for single-income American employees is not the situation for all Americans. Not everyone has an employer who calculates their taxes and pays installments for them throughout the year.

It is common for Americans to have a single job with an employer who calculates and pays their taxes for them. This makes it very easy for the IRS to know exactly how much the taxpayer owes (or is owed) at the end of the year. If it ends up feeling to like this is the same thing as the IRS calculating your taxes for you, however, I'm guessing it's because you forgot that it's actually your employer who's been doing that accounting job for you all along, with each paycheck.