Your router and modem provide different functions. You can not replace one of them with the other, even if the physical media is the same.
You may be able to work around this by finding a modem locally, maybe through OfferUp/Craigslist/etc. Before doing this, though, you'll need to confirm that your ISP even allows BYO equipment - for example, I owned my own modem for my Comcast cable connection, but my AT&T fiber does not allow customer-owned gear.
Cat 5e can handle 1GE at 100m and 10GE up to 45m according to spec. Even 2.5G for a PC is pretty uncommon currently outside of enthusiast hardware, and most consumers don't have 1GE WAN connections, so I think Mom is going to be just fine.