I can see some difficulty, as you point out, for multi-runway airports particularly. Not insurmountable though. Could just broadcast for the one or two most used, and simply not include a component report anytime there wasn't a preferred runway.per the AWAS. And never when "wind calm" is in the report.
Our one runway AWOS typically reports "conditions favor runway three zero" or "Pilot's discretion". Anytime "conditions favor runway three zero" is reported it would report "cross-wind-component for runway three zero is XX knots from the left" or right as case may be. Not do it under other conditions.
Getting the crosswind component simply and clearly audio stated SOMEtimes would be better than what we have now..... which is that even in the clearest and easy to calculate and report case it is NEVER broadcast. Leaving us to make seat of pants guesses.