Complain about it? What's to complain about? Compared to others, the majority of people in the UK have it remarkably easy. As you say, each month the tax is taken off your pay and sent directly to the taxman, and that's it. Zero effort on the part of the person being paid.

If you've got more complicated affairs, you have to do some paperwork, but that'd be also true if the UK had the US system; it's very easy for most.

There's a middle-ground to be found between easy taxes and telling the government everything. The UK government knows monthly pay and takes tax from that accordingly. If one has any other taxable income, or any reason why one should pay less tax, it's up to that individual to tell them. This makes taxes easy (most people never talk to the tax office at all), and maintains privacy.