Why don't we measure spread about the mean by simply averaging x - x, the deviations of individual data values from their mean?

Because this spread will add up to zero

Step-by-step explanation:

Mean defines the center of the data. Some values are below this center and some values are above this center.

For the values which are below the center (i. e. less than the mean), when mean is subtracted from these values, this results in negative numbers.

For the values which are above the center (i. e. greater than the mean), when mean is subtracted from these values, this results in positive numbers.

When these negative and positive numbers are summed together, as it is, they cancel out each other, leaving an answer equal to zero, which would be meaningless. As a spread of 0 would mean all data values are the same, when infact they aren't.

To avoid this, the difference of mean from the data value is squared, so that we can get all positive values, and then these values are added up together to calculate the spread of the data.

Conclusion:

The average of deviations of individual data from the mean will result in answer equal to zero, as a result they are squared first before finding the average.