The 10% early distribution penalty is not waived for a general hardship. You still have to meet the standard penalty exceptions. Although education is a penalty exception for IRAs, it is not for 401k plans.
Reference: https://www.investopedia.com/retirement/relief-401k-hardship-withdrawals/
Furthermore, you did not "already paid all the taxes upfront". What you paid was only withholding, an estimate of the tax and penalty due. No piece of income is taxed individually. All your income, including pension plan distributions, is added together at tax filing time. The tax is calculated on that total (after adjustments, deductions, exemptions and credits). Then withholding is subtracted to determine if you owe more or get a refund.
