How do those other applications obtain the precise value they need without encountering the Internet issue?
They do not use the Internet: they use local (GPS) clocks with internal high-precision clocks for carry-over in case GNSS signal is unavailable:
* https://www.ntp.org/support/vendorlinks/
* https://www.meinbergglobal.com/english/products/ntp-time-ser...
* https://syncworks.com/shop/syncserver-s650-rubidium-090-1520...
Verification and traceability is one reason: it's all very well to claim you're with-in ±x seconds, but your logs may have to say how close you are to the 'legal reality' that is the official time of NIST.
NIST may also send out time via 'private fibre' for certain purposes:
* https://en.wikipedia.org/wiki/White_Rabbit_Project
'Fibre timing' is also important in case of GNSS signal disruption:
* https://www.gpsworld.com/china-finishing-high-precision-grou...
For the most critical applications, you can license a system like Fugro AtomiChron that provides enhanced GNSS timing down to the level of a few nanoseconds. There are a couple of products that do similar things, all based on providing better ephemerides than your receiver can obtain from the satellites themselves.
You can get AtomiChron as an optional subscription with the SparkPNT GPSDO, for instance (https://www.sparkfun.com/sparkpnt-gnss-disciplined-oscillato...).
Then they disperse and use the time as needed.
According to jrronimo, they even had one place splice fiber direct between machines because couplers were causing problems! [1]
Alternate sources include the GPS signal, and the WWVB radio signal, which has a 60kHz carrier wave accurate to less than 1 part in 10^12.
