The entity on which is based the "^ ^ ^" must be specified with "as". If you want the armor stand to fly away, you would have to tp first the armor stand to the player (as it gets the player's rotation) then add "as" :
tp @e[tag=wand,limit=1,sort=nearest] @p (to trigger only at launch)
execute [email protected][type=minecraft:armor_stand,tag=wand] at @s run tp @s ^20 ^0 ^20 facing ~ ~ ~
I don't know if I understood well your question, let me know if I did well!
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