is it true that there is no such thing as an average global temperature?
Obviously not. Defining the statistical calculation is trivial. If you want the average temperature of the planet's surface at a particular instant, just measure the current temperature at every point on the planet and then use calculus to turn that into an average.
Obviously you can't take this down to an infinite number of measurements of infinitesimally-sized plots of land. But you don't have to. The temperature gradient across a fairly large plot of land is satisfactorily continuous. If you only take measurements of every acre (or hectare) or even every square mile (or kilometer), your average isn't going to be off by a ten-thousandth of a degree Fahrenheit (or Celsius).
If you want the average temperature of the planet's surface over a year, then just calculate the average at every moment in time using the methodology described above, and use simple arithmetic to turn that into an average.
You actually can use sensors to get continuous readings, and they'll have built-in computers to do the averaging for you, so you don't have to worry about taking individual readings throughout the year. But even if you do, once again the temperature gradient between two instants half an hour apart will be satisfactorily continous so it won't destroy the precision of your statistic.
Of course being able to describe this methodology is not the same thing as being able to put it into practice. The weather balloons used for global forecasting are something like 500 miles apart, which means they miss
every one of the earth's mountain ranges, which have a profound effect on the weather. If you're measuring the temperature of the planet's surface you won't be able to place them close enough together to put my methodology into practice.
But that's a problem in the application of the methodology, not the methodology itself.
There is indeed "such a thing as an average global temperature," whether you want it for an instant, a day, a year, a decade or a century. You just may not be able to perform the measurement using today's technology, so you won't know what that average is. That's an engineering problem, not a science problem.
does that not make a bit of a nonsense of global warming theories since average temperatures are used to evidence them?
The temperature measurements we have today are adequate to detect global warming. As far apart as our existing sensors are, there are thousands of them. In aggregate, the average temperature they provide for the planet is certainly precise to within a tenth of a degree.
The skepticism directed at the global warming hypothesis is not based on criticism of the accuracy of measurement of the average temperature at any one time, or even the averages tracked over decades. It is based on the assertion that we don't know enough about the forces that influence the earth's weather patterns, in order to be confident that what we're experiencing is a long-term trend rather than short-term. For example, we only dimly understand the effect of sunspots and our predictive model of sunspot cycles has not been tested through enough cycles to earn everyone's trust.
There have been short-term warming-cooling cycles (cycling over a couple of centuries or longer) during historical times. Within the past millennium, there was once a huge network of navigable, liquid-state inland waterways in Scandinavia, and a flourishing vineyard and winemaking industry in England, both of which take more than a few decades of consistent weather to establish.
The most reputable critics of the global warming hypothesis are concerned that we've based a model on observations that have only been collected for a relatively short time, and are using it to predict conditions over a much longer time. Their assertion is that this makes it a very weak hypothesis. Its supporters counter with, "Yeah sure, but what if it's right? We won't have enough time to wait."
Again, this sounds more like a question for engineers than forscientists. It's engineers who figured out how to build airplanes when scientists thought they were impossible with the existing technology.