Here's how it could work, based on a real-world example from another country:
Before each tax season begins, the IRS sends each taxpayer an estimate of how much they are expected to earn that year, based on previous years. They are assigned a tax bracket based on this estimate. If they think the estimate is wrong, they send back a new number and their tax bracket is adjusted accordingly. The taxpayer can also give estimates of their deductions at this point, for a more accurate estimate of the tax bracket.
Employees withhold taxes from wages before payment, based on the tax bracket each employee is in. They report the wages paid and pay the IRS the withheld amount.
Investment funds etc. are required to report gains and losses for each of their clients to the IRS.
At the end of the year, the IRS sums up the income they know about from these reports, compare the sum due to the sum already paid from withheld taxes and send the taxpayer a summary. If the taxpayer has no other income to report, and has no complaints about the data the IRS already has, they don't need to file anything further. The report comes with a bill for taxes remaining, or if too much has been paid, a summary of how much will be returned.
For the majority of taxpayers, that is, those whose source of income is wages from employment or government benefits, all this takes maybe 10 minutes a year to handle. Some people still need to file, but they're a minority.