I think in this case the most likely thing is that the 172.16.0.0 network already existed and then they added the router, not really knowing what was going on and rather than put one in the others range they ended up with the above arrangement. That or possibly the system was originally on a no-nat broadband service so they could only have one address on the "public" NIC.
This is my thoughts exactly - the number of "technicians" who operate a "dont understand it + its working = leave it alone" isn't always a good thing ... before you know it: http://sysadminday.com/images/horror/Backbone.jpg
Is this missing something obvious?
Not that I can see...believe me I have seen this and worse many times (including the recent new client who has a "private" subnet starting at 11.22.33.0)
Before "internet access" really became common, there where lots of places which that did non standard ip addresses - I think a lot of people had upgraded from various network architectures (Netboi, ethernet, token ring etc) and without looking into any specs of IP, they understood the principles of subnets and just picked random numbers for their private networks.
Oh the difficulty some of these manufacturers had when they decided to give connectivity to all workstations, and realized that basically every machine (industrial machines as well as computers) had to be changed from their static addresses because they had just made 1/256th (or more) of the internet disappear from their sights.
James