That's the cynical take, but as far as I can tell it is far from universally true. You just hear about all the bad cases. Not to mention, the the way it pulls on policy can be much more subtle than getting bills passed.
They can do the old fashioned make things better to make the voters happy and get reelected. In London the mayor has brought in a lot of anti pollution measures and although the tabloids and right wing media go on about how awful he is, he gets reelected, including my voting.