The logic was that since the cable modem is often equipped with a cheaper processor as it is supplied by your provider, that it often chokes even crashes when you start a torrent or something else with a lot of connections.
The cable modem's only job, if it's a typical cable modem, is to accept DOCSIS traffic from the cable side and bridge it to Ethernet traffic on the consumer side. It has a DHCP client since that's how ISPs typically hand out IPs and a small webserver for diagnostic/status purposes. It's not typically modifying the traffic or doing any processing to the packets that come in and out of it.
Cable modems that support VoIP, such as the Arris MTAs that Comcast gives out (or have given out) to support phone service, handle more.
To troubleshoot for sure, you need to take your router out of the loop. Connect a PC directly to your cable modem, without a router in the middle, and then perform some sustained high-traffic activity. If your cable modem is indeed crashing due to excessive traffic, that's quite unusual and you should replace it or contact your ISP to replace it.
Typically the router is at fault when this happens, since generally, to support firewalling and such, every IP packet needs to be examined by its CPU and consumes memory, etc. The cable modem doesn't do anything to the traffic so the amount of traffic passed through it shouldn't affect CPU usage.