I only said you ought not to arithmetically mean vector quantities, not that it is impossible!
Vectors have both size and direction, and 25kt tailwind then 25kt headwind does not mean to 25 + 25 = 50, divided by 2 equals 25. You solve vectors graphically all the time on the nav plot on the wizzwheel, and if you wanted an accurate mean wind you could stick them all on the wizz. Another way would be to first factor the winds to a component along a chosen track for each spot then add up and divide by the number of spots to give a wind vector along the track. Next repeat for across track. Finally, sum the cross track and along track vectors to find the average, or mean, wind.
This is not worth the effort, so there are two easy ways out. One is to gash it, and guess the mean wind. However, if the wind directions to be meaned are nearly the same you can just add up all the speeds on the spots and divide by the number of spots to get a mean speed that is good enough for government work. The mean direction, by definition if we are using this method, is roughly the same as the arithmetic mean of all the directions.
Because, as I said, on the 214 the spots are close together geographically, the adjacent winds are similar, the arithmetic mean works. On the big high level charts the winds can vary in direction 180deg, spot to spot, so it doesn't work. We used to teach the gash method for the ATPL, but I agree, mean winds did not often come up.
However, back to the begining, if you need an absolutely accurate mean wind there is no substitute for a full vector calculation.
Dick W