All archery competitions, no matter if on a getaway, community lot or at home, work for a time but do not complete. Just before it ends the sims just loose their task and walk away without any outcom...
I have a child sim and I first chose to do a competition with his imaginary friend. My sim did his part but the imaginary friend seemed clueless about any part he would play. So I had him do it with the butler. The butler did hers then took off. Child's queue just said 'wait'. Same if I tried with him starting one with his mom.
I finally tried again with the butler, but ran out of time where he had to head to school (I thought two hours would be enough to complete!). Plus another sim can over and butted in. However, at least this time they took turns... and to my surprise, despite all this chaos, that last 'competition' finally counted!
Edit: I had this same child sim also do the archery competition with the butler for the three 'win a competition'. The three competitions did work okay, and I got the notification that the child won each of them... however the check-off are in the aspiration did not check off for any of these. I even tried saving and exiting to manage worlds, then to exit the game, and then re-entering the game, and neither of these got the item to checkoff... so in the end I had to give up and enter the "aspirations.complete_current_milestone" cheat to complete that last part of the aspiration, which was anti-climactic, frustrating, and disappointing (that's why it's greyed out in this screenshot). I also noticed that 'formative moment' (Into Archery as a Kid) for this sim child that had to do with archery competition did complete for him amidst all that. I have no mods/cc, my game is fully updated, I have all packs, and play on Mac.