It's handled by a separate layer, yes, but that doesn't mean that the routing layer magically disappears. It's still there, and in fact it's still required to f

It's handled by a separate layer, yes, but that doesn't mean that the routing layer magically disappears. It's still there, and in fact it's still required to forward every single packet that goes through the router. NAT is just simply an extra layer on top that sometimes changes the addresses in the src or dst fields of some packets. Dropping packets not sent to a valid target is called firewalling. NAT doesn't decide what's valid and what isn't; all it does is detect packets which correlate to known connections and rewrite some of the headers on those packets. Home routers, like all routers, do a lot of routing. They sit between two networks and they forward packets between them, which is pretty much the definition of a router. Having some header rewriting going on doesn't change that.