There are really two things here:

1) The typical "NAT" is a "PAT", which requires state for mapping traffic flow. An implementation is going to look very similar to a stateful firewall, by necessity, not accident.

2) The use of RFC-1918 addresses behind the router / NAT box provides some protection. If your address is not globally reachable, it's much harder to reach any hosts behind it. Yes, a poor implementation might allow direct routing from adjacent networks, like your ISP or neighbor. But that still isolates you from 99.99% of the internet.