> No, the side effect of NAT is that outbound connections made from your network look like they come from the router's WAN IP.

That's the primary function of NAT, not a side effect.

> It doesn't filter incoming traffic.

Of course it does, it drops any incoming traffic for which it cannot find a corresponding connection. How is this not a filter?

I know that internally these two are vastly different. The reality is that NAT is used as protection for millions of home networks.