The withholding is not a calculation of actual taxes owed. E.g. it doesn't even have a way to account for spousal income for married couples.
You can then retroactively go in and modify that by doing a "Arbeitnehmerveranlagung" where you can claim modifications to your automatic tax filings.
This is a different process for when you are self employed and you actually file taxes from "scratch" which however is also a significantly simplified process.
People with various sorts of cash businesses that don't appear at any of the classic touch points are few and far between, and probably don't report anyway.
And for the remaining people who, say, want to record an expense to offset a capital gain (like a remodel that increases the sale price of their home): they have an incentive to let the tax authorities know, since it reduces taxes. I doubt many people would do this more than once or twice in their lives unless they ran a real estate business.
The systems described by the article seem to be using the data submitted by employers and financial institutions to generate a tax return, which the tax payer verifies or adds to (in the case of self-reported income). For the majority, filing taxes would either be a simple confirmation that the government has complete data or vastly simplified because a great deal of the work has already been done for them.