I thought that the experience of developing the GPS network (where all the sattelites and ground stations must be time co-ordinated to within 10+ decimal places in seconds) would have helped them with this missile system.

A GPS satellite travels much faster than a missile, and is accurate to within meters, so the tolerances are much higher (and the system was designed prior to patriot).

There is a definite 'not invented here' syndrome amongst defense contractors - I doubt they share any information, research or solutions amongst one other, which means the US tax payer foots the bill each time one of these contractors must independently develop and implement a system that has likely already been built in another part of defense.