I don't think that works. A router should decrement the TTL of the frame, and thus showing that there is a router between the host device. The linux default is 64 and windows is 128 IIRC, so you can easily deduce the OS just from looking at the TTL. This can tell you whether an ipv4 device is directly connected. From there, you just need to look at IP ID in the packet and figure out which ones are increasing independently to determine individual devices behind the NAT.

So, no. NAT gives you 0 privacy.