From what i can understand, this separation of concerns is a safer option
Besides, signing doesn't necessarily help - the easiest way to GPS spoof is simply to re-broadcast the signals received at a nearby point at higher power, so the victim receiver thinks it's at that point.
There is recent work on doing this: https://web.stanford.edu/group/scpnt/gpslab/pubs/papers/Lo_I...
replay attack does make sense, but isn't it identified uniquely by time? i mean the content of the payload is basically very accurate time. if a device notices the time doesn't change - it can detect spoofing.
Receiving the same signal over and over would be suspicious, but unless you have some other method to determine time and position, what are you going to do?