David,
I have likewise not found any information about how the USNO calculate the values, or how accurate they are, though I have admittedly not looked very hard. I would expect they’re pretty good, though. I’m sure somebody like Jean Meeus would have noticed by now if they were publishing times that were wrong!
I think that if the times are computed using the most accurate data available, one-minute precision should be easily achievable. Both the Earth’s orbit and its rotation are very well characterised. Its rotation is measured by monitoring distant quasars with radio interferometers, which can achieve micro-arcsecond precision. That precision is required by Gaia, for example, which is doing astrometry at micro-arcsecond precision in a coordinate system defined by the Earth’s rotation axis.
The Earth’s orbit is very well characterised from spacecraft tracking work (Nick James’ day job). The idea is to track the position of spacecraft by measuring the time taken for telemetry to reach them from ground stations. In extreme cases (e.g. Rosetta) this has been done to one-metre precision or better. That requires you to know the position of each ground antenna to one-metre precision, which is no mean feat as that’s below the scale of seismic oscillations in the Earth’s crust…
So my guess would be that the Earth’s orbit is known to about one-metre precision, and the position of the celestial poles is known to micro-arcsecond precision. Of course, that’s not to say the USNO used an ephemeris that was that good. I would think it’s very likely they used some publicly-available ephemeris of lower accuracy.