The "Pay-As-You-Earn" system was introduced in Australia and the UK during WW2 lol. The US has the same system its just been implemented extremely poorly
It's not that it's implemented poorly. It's that 1) the federal government collects multiple kinds of taxes out out of your paycheck. Social security, medicare and regular income tax are different. Income tax rates are progressive while the payroll taxes are flat but capped at a certain amount. 2) Additionally forty-two states have income taxes as do dozens of local municipalities. Consider that they also don't treat residents and non residents alike.
Consider a couple who live in New York City, where one partner works in Connecticut and the other works in New Jersey. There are potentially 6 different entities collecting income tax there.
That being said - fully agree - tax filing software should be free for everyone.
Furthermore, a number of states are starting to require tax returns to be filed even for short durations of working in the state for business purposes.