Could you describe a scenario where this could be the case? What kind of things would abusive tax prep software do? Would it lie to get people to overpay? Why would the government take that approach instead of just levying directly from people's accounts and/or reducing their refunds?
This is a fundamental misunderstanding of the situation. The tax law is what it Congress decides it is. The law depends on taxpayers to report their incomes as well as deductions and other facets of their financial lives (e.g. household size, addresses, etc.).
A government-run tax prep site couldn't necessarily pull in all of your deductions, that's correct. But it also couldn't necessarily pull in all your income (especially business income, offshore income, etc.). It's mostly neutral in that respect.
In this case, it is mostly one sided
Roughly 90% of Americans file using the standard deduction ($13k for a single person, $26k for a married couple filing jointly). Median household income is ~$67k, so the median household is not itemizing.
The remaining 10% of people would not be bound to use the free public system.
That said, Trumps tax law greatly increased the standard deduction so it's not as big of an issue any more.
As of 2019, only ~11% of filers itemized. It would be a good idea to have a public website where 90% of filers could do their taxes for free.
Personally I hate the fact we have so many deductions. I'd love a simplified tax code but I think that would be a tough sell.