The Danish system is only super simple if you are a salaried employee. Once you run your own business it’s generally as difficult as the American system.
You have to have your books in order anyway, and tax filing is usually as simple as copying over some numbers from the bookkeeping software and then Skatteverket (Swedish IRS) will suggest what is probably the best deductions and periodizations available for you. For my one man consulting business it takes less than an hour, and requires nothing more than having my accounting in order.
But if you are self employed, you have to hire an accountant. It is not a legal obligation but there are so many things that can go wrong that you are putting your business at risk if you don't.