The manufacturer tried using two instruments to measure the shape of the mirror. One instrument said that the mirror had unacceptable error. The other said it was perfect. The latter instrument was designed to be more precise, so they went with that and ignored the measurements from the other null corrector which disagreed.
The problem was, the more precise null corrector was the same one that was used to help construct the mirror originally. All it could verify was that the mirror was built to the same specs as the null corrector. No one realized that the "better" null corrector itself was off until it was too late.
Thinking about it more, I've never tried asking ATC for an altitude check specifically from primary radar. It may well be that the system isn't set up for it, even if it's technically computable to some degree. When I'm talking coarse, I mean an altitude spread at range that may be measured in miles.
Surveillance radars have narrow beam widths in azimuth, but broad in altitude to allow them to quickly survey around them and get good enough awareness of where everyone is. There's always a trade-off between resolution and volume--some military tactical and approach radars have very precise azimuth and altitude capability, at the cost of scan volume/time. Mode C (and now, arguably, ADS-B) was the solution to this problem, allowing ATC radar to have both good position and altitude precision with an acceptable refresh rate, at a non-prohibitive cost.
You'd be surprised--unless your hypothetical binocular observer was really handy with stadiametric ranging, I imagine they'd be nowhere near a mile. They might be able to get a decent azimuth and elevation on a single aircraft, but that's a much different problem than determining position near-simultaneously for a hundred.
In the case of this flight, maintenance tape covered the static ports, rendering all instruments that use the static pressure system, significantly in error. Although there's more than one port, and thus redundancy, the maintenance operation covered all of the ports.
The pilots suspected their altitude indicator was wrong, so they asked air traffic control to confirm their altitude. They did and it agreed with their own readings so they ignored the subsequent indications that they were going to crash into the ocean, which is unfortunately what happened.
Neither the pilots nor air traffic control apparently knew that both readings were supplied by the same source: the plane’s transponder.
I often think of this when I hear people say that multiple sources indicate something strange. These systems are often intertwined and networked in unintuitive ways and just because something appears on multiple panes of glass does not necessarily mean it’s been independently confirmed.