Do measurements really deviate wildly for different well-designed upstream components under non-ideal loads?\
Not at all. As usual, you have to design the measurement to consider actual use.
It doesn't seem like they're measuring the right things.
Exactly. The problem is that modern headphones and earphones often have very low impedances (for traditional headphones), such as 8 or 16 ohms. Amplifiers that drive them need to (and often do) provide relatively low source impedances, generally well under 1 ohm. This sort of thing is well known to designers and testers of portable digital players. For example many vendors have been supplying portable digital players such as later generation iPods and the Sansa Fuze that provide source impedances of well under 1 ohm.
The Benchmark Media white paper http://test.benchmarkmedia.com/discuss/sites/default/files/Headphone-Amplifier-Performance-Part-1.pdf
mentions the Sony MDR-V6. Here is its impedance curve from http://www.innerfidelity.com/images/SonyMDRV6.pdf
The lowest point of this curve is 60 ohms, but it goes as high as 120 ohms at 40 Hz. This means that a headphone amp that provides a source impedance of 60 ohms is for example going to produce excess output at 40 Hz.
You can do an adequate job of measuring the source impedance of this headphone amp with a 60 ohm resistor, but you have to do the appropriate calculations:
A common rule of thumb is that an amplifier needs to provide a source impedance of no more than 1/50th to as little 1/100th or less of the load impedance, when the load impedance varies with frequency over the operational range. This ratio is the same as the inverse of damping factor.
Many headphone amplifiers have an output impedance in the range of 0.5 - 50 Ohms. The 1996 IEC 61938 standard recommended an output impedance of 120 Ohms, but in practice this is rarely used and not recommended with modern headphones. High output impedance can result in frequency response fluctuations, due to varying load impedance at different frequencies. In 2008 Stereophile Magazine published an article that showed that a 120-Ohm output impedance could cause a 5-dB error in frequency response with certain types of headphones. However, the author of the article also states: ″The ramifications for subjective assessment of headphones are more troublesome because it is usually unclear what assumptions the manufacturer has made regarding source impedance. ″ 
More importantly, low output impedance can reduce distortion by improving the control that the source has over the transducer. This is often expressed as damping factor, with higher damping factors greatly reducing distortion. One company shows a 45 dB improvement in THD+N at 30 Hz for their low-impedance amplifier compared to a 30-ohm amplifier. For example, a 32 Ω headphone driven by a headphone amp with a <1 Ω output impedance would have a damping factor of >32, whereas the same headphone driven with an iPod Touch (7 Ω output impedance)  would have a damping factor of just 4.6. If the 120 ohms recommendation is applied, the damping factor would be an unacceptably low 0.26 and consequently distortion would be significantly higher.