I've since had a look around other towns and found plenty of examples where it's mapped as you've done it. The difference is that generally (not always) only one object is named. Removing the name from the area feels like the right solution in this situation where the linear characteristic is slightly more important than the area (i.e it still feels more like a thoroughfare than a square). I think the Broadmead ways would benefit from the same treatment although not sure whether to tag the way or area in that case.