Issue 1: Could be the physical board itself. Depending on where it was purchased, it could be poorly-designed, poorly-manufactured, using counterfeit parts, etc. This board looks to me to be a low-cost Asia-sourced unit, built in large quantity and sold cheaply - the PCB looks thin, there's no manufacturer label or serial number, etc. It is my experience that many low-cost direct-sell vendors found on places like eBay, Wish, etc. have issues with product quality - the items don't fully meet spec or don't last a long time. Some will gladly sell you a replacement, some will send you a new one for free. Some won't return your messages. There are larger domestic suppliers (Digi-key, Mouser, Farnell, etc.) which demand better quality and may offer support if you end up with a lemon - at a higher purchase price, however.
Issue 2: Could be the application - was there input protection installed? You didn't mention having things like a fuse, input polarity protection or inrush protection. A fuse disconnects the source from the load if something bad happens. Input polarity protection prevents a reverse-connected battery from doing anything. Inrush limiting protects the input caps from seeing damage due to high di/dt from the battery (which can deliver a lot more current than the wall-wart adapter which would have been supplied with the router!). You also could consider a TVS across the input of the buck to protect the cap from induced voltage stress if the wires are long and there's a sudden current interruption.
Issue 3: Could be the design. Example - limited power handling of the LM2596S. Because the integrated power switch isn't mounted on anything meaningful (as far as power dissipation is concerned) it isn't going to survive at peak power for very long. It's a 3A rated part but the device needs to be mounted on some serious copper to be able to deliver that. Did the supplier offer you any test data or qualification results showing what the device is, and if it needs external cooling?
So, consider input protection and inrush limiting at the very least. Also consider investing in a scope (even an old, used one is useful) and start looking at the waveforms if you really want to know what's going on. Check device temperatures when the circuit is running steady-state - is anything getting hot prior to failure? Are there any high voltages you can't explain? Excessive ringing? Your intended application, sadly, is not "plug and play" - a buck regulator in this sort of package isn't like a wall-wart adapter that you can just plug in and use. It takes a certain amount of knowledge to diagnose and resolve why there may be issues.