The crux of the question is in the title. I am not familiar with Network Analyst (only played around once or twice), and I am wondering if this is a capability (and in this case, 'capability' can be either out-of-the-box tool or something ascertained by cobbling together a suite of tools).
I am attempting to identify polygons (in this case representing subdivisions) that only have a single mode of ingress/egress. Can this be done in Network Analyst?
An intersection would be key, but use the roads with a polyline version of the subdivisions. You will get a point at every entry/exit point. You can Summarize on polyline's objectid field. Any count equal to 1 only has one entry point
I think this is a good place to start, so Network Analyst won't even be required.
Due to the nature of the data and what I'm examining this will end up giving false positives, as there are roads that might show up in a centerlines layer but are not, in fact, viable modes of egress, but it will be a good place to start.