That's not the case at all. You could disable their NAT and they wouldn't lose any protection whatsoever.
Any inbound connection that would have worked before you turned it off will still work afterwards, and any that wouldn't have worked before will still not work afterwards.
Maybe this is the reason for some of the disagreement. I am focusing on what is installed at 99% of user installations (PAT). I would agree with the comments that a 1-to-1 NAT offers no EXTRA security.
Connections to the router's IP address go to the router, but you need to consider what happens to connections that go to IP addresses on the network behind the router too.