If your network (i.e., just the router and devices) works without an internet connection to the modem then that is a good thing. If the network performance degrades when the modem is connected to the router then that narrows the possibilities.
Unfortunately many ISP techs are poorly trained, over-worked, and are often abused by customers who take out their frustrations on the tech.
Does not take much to make a cable or connection fail and intermittent connections are the worst.
When Cox came on site did you notice or did the tech mention doing any sort of cable testing: i.e., measure signal strengths and continuity?
What modem make and model is installed? You may be able to access that modem via a browser interface to customize the configuration, look at logs, do some troubleshooting, and even check incoming signal strengths. Knowing the modem make and model may lead to the applicable User Guide/Manual. Then we can see what else may be learned via the modem itself and its configuration. I will note that it is beginning to seem as if such functions are being "disabled" more and more by the ISPs and even the device manufacturers. Savy end-users are not in the ISP's best interests....
Inspect as much of the incoming coax cable as you can and as safely as you can - do not climb poles or roofs. Look for dangling, unconnected coax cables, unterminated splitters or coax outlets. They should not be just covered - they should be terminated with a special cap. Go off-premise if necessary. See if you can find any unexplained splitters - maybe someone is stealing service.
Do your cables go underground? In my area it is not at all uncommon to see the heavy duty bright orange coax cables strung along streets, sidewalks, yards, and houses. Temporary connections until some third party contractor shows up to bury a new underground cable. Rarely more than shovel deep. Wedge up the sod and drop in the cable.... Old cables are not even removed.
Anyway if you are not already doing so, keep a written record of events with Cox. Keep calling, write down names, dates, times, and honestly note what transpired. Keep calling. Remember that the Customer Service Rep. is at the bottom of the process and could well lose his or her job if they depart from the script. Agree to do the post call/visit evaluations when asked but keep the evaluation truthful and objective. Vote down the company - not the person (unless that person is demonstrately unprofessional). Many Reps are not well-trained at all and must follow some script. In all fairness though, scripts can provide a methodical approach that will resolve the majority of problems. Very annoying to be asked (I know) "is it plugged in" type questions. But, (again I know) sometimes it is not. 🙂
More immediately - the objective is to see if more can be learned via the modem and maybe even the router's logs.