@parados,
In regard to this argument, most of the time the average of the high and low will be a decent measure of the temperature, and if all the records are done in that fashion, that would seem to be a reasonable measure. However, the average of the high and low obviously would not yield the most perfectly accurate result.
For example, one day has a high and low of 50 and 10. The next day, a chinook wind in a mountainous area could occur during the night, taking a temperature of 0 up to 35 or 40, and perhaps a high of 48. It is highly possible and even likely that although the high and low were 48 and 0, if the temperature had been monitored throughout the 24 hour period, the average could very well be higher than the 30 from the previous day. This is but one of countless examples of why the average of a high and low may not yield the most accurate result for average temperature. Obviously, a higher number of readings, such as every hour or even every minute, or even every second if the equipment is capable of doing it, the average would become more and more accurate.
I have not followed the entire argument over this point here, but is it feasible to monitor the temperature throughout a 24 hour period for every station around the globe, in other words do we have the technical capability for doing this in a reliable way, and even if we did, I imagine we would have no similar set of historical records that could be compared to it. I think consistency in measurement is at least as important as the perfection of accuracy that can be attained.