This research studies how populations depend on road networks for access to health care during crises or natural disasters. So far, most researchers rather studied the accessibility of the whole network or the cost of network disruptions in general, rather than as a function of the accessibility of specific priority destinations like hospitals. Even short delays in accessing healthcare can have significant adverse consequences. We carry out a comprehensive stress test of the entire Austrian road network from this perspective. We simplify the whole network into one consisting of what we call accessibility corridors, deleting single corridors to evaluate the change in accessibility of populations to healthcare. The data created by our stress test was used to generate an importance ranking of the corridors. The findings suggest that certain road segments and corridors are orders of magnitude more important in terms of access to hospitals than the typical one. Our method also highlights vulnerable municipalities and hospitals who may experience demand surges as populations are cut off from their usual nearest hospitals. Even though the skewed importance of some corridors highlights vulnerabilities, they provide policymakers with a clear agenda.