not the sim card, its something to do with the sim IC , if it is an i337 or m919 i have yet to find schematic, when i get that job at work I don't take it on as I don't know any solution. Look at a new device as even repairing it at this point if you could find someone who REALLY knew what to replace and could do it I imagine they would charge $100-$150 and at that point you might want to pass it on to someone who could use it as an iPod or just sell it for the screen and what not.
yes it is something that happens to some but not all s4 and is a common issue. If you google no sim s4 you will see lots of talk about it. If you do have a micro sim NOT a nano sim with an adapter that has helped some of my customers. I understand about the waste aspect. It is a shame this is still a great phone. Great fit in my hand and decent camera, etc. you're welcome, and all the best!