I've always wondered - if special and general relativity weren't known, could the GPS engineers have just measured the error and empirically accounted for it? I don't know relativity or GPS or engineering well enough to answer.
It's possible, but it seems more likely that the first tests would have shown "weird timing errors we couldnt diagnose" and then the project never got a follow up.
Yeah, absolutely. You'd be surprised how much high-precision engineering absolutely relies on "this but you have to mill a tiny amount off the end because if you do the maths it comes out ever so slightly too big, and then adjust on test".
Yes, the clock slippage is a constant factor, entirely predictable. If made public it would probably have catalyzed the discovery of special relativity soon afterwards, but then people would’ve been super confused because its predictions still wouldn’t match the observations and indeed would widen the discrepancy rather than narrowing it!