Personally, I'd put the old router back in the box and bury it in the basement. Although, to me, the culprit looks more like an issue with the website itself being infected I wouldn't "trust" your old router on your network until either the issue reoccurs with the new router (solidifying the infected website theory) or you find some other cause. I suspect the issue is likely to come back either due to the website, as mentioned, or some other infection on your network. You don't seem convinced that this isn't malware of some sort, but it sure looks like it to me. If it does come back, then you may as well use your old router as a spare or AP.