I hope we get the GPS III constellation up and running before any catastrophic LEO cascades. Having a self sustaining common clock system that can run for centuries has a very high scientific value (and really, just a huge value for any industrial, commercial, and consumer applications).
From what information are you assuming these will run for centuries? Most sats have a design life of like 10 to 25 years max. Also keep in mind that station-keeping and correction data is mandatory to keep the accuracy in check. I think I read somewhere that after 180 days of un-corrected drift the GPS would be able to get you within a few km of your target - not meters/feet.
I was told by someone who follows theses things more closely than me that block III GPS would have an inter satellite network that could share clock information independently of ground control. While looking into it the closest I could find was references to OCX, which is ground based controls. So maybe we're not there yet idk.
LEO cascades would not affect GPS in any way, as GPS satellites are in much higher orbits.
LEO cascades would affect our ability to launch anything into orbit.