There are notable exceptions, like senior citizens - which is why we haven't done anything about the unfunded expediture on the Medicare prescription coverage.
Most of the "failures" you see are the result of a dysfunctional political system. For example, the Post Office was looking to branch out into services like bill payments, identity verification, banking, etc but Congress forbid them from doing it. Not to mention the crazy requirements Congress set for pre-funding their retirees. No private company has to do that - most of them are pushing their funding obligations off to the future. Underfunding retirement isn't a good idea by any means, but the playing field is most definitely not level here, both in terms of responsibilities or ability to pivot.
It's wildly unfair to blast government agencies for not innovating. They don't have a CEO or a board, they have Congress calling the shots. Of course they aren't agile. And even worse, half of Congress thinks the entire concept of government services is illegitimate and is actively engaged in sabotaging and defunding those services. There's not really a parallel in the private world where a CEO is destroying a company because he thinks it shouldn't exist.
That's part of the problem. Government programs don't need to fix or eliminate problems. Once they've been launched, it's virtually impossible to get rid of them. We're stuck with the War On Drugs, the TSA, ethanol subsidies, and on and on and on.
Fixing bad programs is very difficult, and once we've set something in motion, even if it was a really bad idea, it's all but impossible to pull it back.
EDIT: freudian slip - I typed "government problems" in my first para, where I should have written (and have changed to) "government programs".