Isn't that just what the bad telcos are trying to claim? Of course your router needs to be able to route/NAT with sufficient throughput, but it shouldn't matter if you have your own router or a Zwangsrouter (router forced upon you by the telco as part of the service).
Thing is that the NAT performance of a router is not simply X number of Mbits. Its common for a router to show great full link speed throughput in easy to route scenarios, but outright crash when presented with a particularly difficult routing scenario.
An easy routing scenario is no firewall rules and 1 client, sending back and forth max sized packets at 1 server. Lots of cheep junk routers will do full max link speed on that. When you have 10 clients on a network talking to 1000s of different servers, constantly opening and closing TCP connections while sending large bursts of tiny packets will generally bring most non professional routers to there knees, making them drop to something like 10% of the max link throughput.
One particularly nasty thing to do to a router is use a high throughput port scanner software like nmap and just let it rip without any speed limits. It can send millions of TCP requests trough a 1Gbit link. This is so violent on cheep routers that it causes some to crash and not recover until a reboot. I heard of a guy with a fast internet connection getting a call from there ISP when using it. He told them that he is port scanning and the IT tech on the other side was fine with it, but asked him to turn down the packet rate because he overloaded the ISPs backend networking equipment.
And this is the problem with a lot of routers that ISPs give out. They work fine in the ideal scenario but often fall on there face in a more demanding one. Luckily my ISP is nice enough to flash my modem with a special "dumb modem only" firmware with all of the routing functionality stripped out, while i use my own more enterprise-ish grade MikroTik router to do my routing. It survives the worst case nmap scenario just fine with web browsing remaining functional for all the other clients, but the CPU and memory usage graphs on that router show that it is really really working hard(as in >100x typical resource usage)