What precision was common when surveying with a Solar Compass?
Obviously magnetic compass it was common to report to the nearest 1/4 degree but I'm not sure about solar compass, if they could/would report a finer precision than that (such as nearest 1 minute).
Properly used, the typical statements of accuracy that I see are 1 to 3 minutes. Obviously depends on adjustment of instrument, accuracy of latitude, proper settings on the instrument, time of day (altitude of sun), etc. But if all is done properly, expect 1 to 3 minutes.
My impression is that Dave Ingram is quoting the ultimate achievable accuracy, and most uses would not have been that accurate.
Readings taken too near the middle of the day would have a poorer celestial triangle and not achieve that. And the guy is certainly going to keep working during most of those hours.
I see that in a treaty line survey in 1843, which was a fairly early use of the solar compass by others besides Burt, that he recorded everything to the nearest 5'. He also didn't run the line with the solar instrument, but rather checked the declination or "variation" for his magnetic compass every few miles.
Right you are as to best case. And as you point out, time of day is critical.
But most solar instruments has a magnetic compass on them so that you could keep working during bad times of day or cloudy conditions.
I've got a chart around here someplace - let me see if I can find it.
Found the chart. This will give you some perspective:

Do you have this book? I don't but have been wanting one.