Wouldn't a very rough location suffice, i.e. something that could be either manually set (for general aviation and shorter distances), derived from a VOR station identifier or similar, or just estimated via dead reckoning/an INS?

In other words, if you don't even have a rough idea of where you are, what good will a magnetic heading do?

Rough yeah - but I think it does change by integer degrees from actual map sheet to map sheet, so you could probably drift a couple of degrees without knowing it within an hour or so of flying.