Exactly. Archery doesn't provide strength training if you have to do strength training to do archery. If it would be good at strength training than archers wouldn't need separate strength training.

That's incorrect. Every sport requires additional strength training in order to perform at a high level. Even in strength sports, supplemental strength training is required beyond practicing the sport itself. This doesn't mean that the sport itself doesn't provide a strength adaptation response. Yoga counts as strength training for the sufficiently weak.

In Olympic weightlifting, the contested lifts are the snatch and the clean and jerk. Even minimalistic weightlifting programming involves squatting, and most programs include pressing, rows, deadlifting, and other strength work as well.

Powerlifting is a much simpler sport, testing only the squat, bench press, and deadlift for one repetition. Just practicing the sport would involve doing single reps with squats, bench presses, and deadlifts. Virtually no successful powerlifters train this way. Basically all of them do multiple repetitions on the main lifts, and the majority do other exercises as well.

That still doesn't seem right to me, but I should point out that a good motivation to do a thing is as valuable as the thing itself, if otherwise you wouldn't.