Celestial navigation requires an accurate clock and an almanac that needs to be updated. It is completely impractical for non-military applications.
Does the almanac need updating? I regularly use plate solving with my telescope and I've never had to update it's stars database
Is that really that impractical?
With a quick calculation 50 ms time offset would mean about 20 meter difference. You can get that accuracy with NTP over residential internet and a better TCXO could hold that for a day. You could sync clocks and download the almanac at each airport. The almanac could be digitally signed and NTP has an option for authentication, too. That's about 10 USD of hardware (plus what you need for image capture and processing).
I'm pretty sure with dedicated hardware and better solutions you could get much better accuracy.