> In the most commonly (almost exclusively, in case of end users) deployed setups (call them NAT Masquerade, NAT Hide, NAT Overload, NAT Many-to-One, PAT, whatever) it does block incoming connections, only allowing replies to internally-originated traffic, and to do that it tracks the state of all traffic.

Yes, but that's because it's co-located with a stateful firewall, sharing the same connection tracking state. Without that firewall, if a device on the WAN side sends to your router a packet with destination address on the LAN side, your router will route that packet to the LAN, even though it's not a reply to anything sent from the LAN to the WAN. That is: it's a misconception that port mapping NAT blocks incoming connections; port mapping NAT only affects outgoing connections and replies to them.