Assuming it's not a 1-1 NAT it does make for a functional unidirectional firewall.
That's like saying a router and firewall are the same thing. NAT appears to be a "firewall" because it's usually deployed with one. NAT itself has no filtering functions the way you're describing.
Now, a pure router in the sense of simply offering a gateway to another subnet
A "pure" router, as you put it, understands upstream subnets and routing tables. NAT does not, and is usually overlayed on top of an existing routing function.
You can set up NAT between two subnets as an experiment with no iptables and it will do its job.