We've noticed that uniform circular motion and a spherical shape for the Earth work well as a first approximation. But as one gets more precise in one's measurements one notices smaller changes such as small changes in the length of the day over the year and the effect of the flattening of the Earth. Vertical deflection affects the determination of one's position to one minute of arc accuracy. To measure changes over long periods of time high precision is not necessary. But for the measurement of rapid changes over short periods of time more precise measurements are. So the answer to the question of whether or not we need high precision depends what we are doing.
In the case of global warming changes are more noticable over long periods of time. Currently we are concerned about the effect of greenhouse gases on our environment and we need accuracy to fractions of a degree in temperature to measure this. If we are to be consistent we also need to take into consideration the small changes of other effects over time in order to get a better picture of what is happening.
One also needs to take into consideration the level of education one is working at. For the high school and elementary levels high precision is not necessary for the introductions of our classes. But at the university level one expects more expertise and a more detailed discussion is necessary. Context is important.
So, does the general public need to concern itself with minutiae?