However, my limited knowledge of the problem precludes me from understanding if a simple 24 hour re-sync would address the underlying problem.
From what I do understand the accuracy/precision of the GPS system would be affected. I'm also inclined to believe that even with a regular re-synch, the overall usefulness of the system would be affected. As I don't believe the re-synch itself would be exempt from the underlying problem.
What I do know is that given our better conceptual framework we're able to leverage very precise location information; this leads to a more useful and productive GPS system than we would have otherwise.
But I picture it like this, in my very naive way.
1) launch satellites
2) after some time, engineers notice the clocks keep running behind/ahead (I always forget)
3) engineers debate, and have no clue what's happening
4) engineers build fix to re-synch the clocks every 24 hours or so
5) gps system works fine, engineers remain puzzled
And then of course
6) after years of puzzlement, some kid figures out what's going on