"but if you died part way thru a month, you might have an extra days' pension you are not entitled to"
No you wouldn't because it's paid in arrears. However, if you died (say) 10 days before the payment was made and the Pensions people were not told in time to stop the payment, your executors would have to pay back 10 days' pension. This would apply whether the payments were made every four weeks or every calendar month.
Almost everybody's bills these days are calendar monthly. Those relying solely on a State Pension must find it very difficult to manage their affairs when the date of their bills and the date of their pension payment can be up to two weeks apart.