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"at all levels"
If "at all levels," you mean sub-varsity, not just NCAA, NFHS and FIBA, I doubt it would save time in 2-ref sub-varsity games.
In the old days, with two officials, the T would step into the lane with the ball, announce the number of shots, turn and hand the ball to the shooter, then step out and back. If I recall correctly, it was precisely to speed things up that the mechanic was changed to have the L administer FTs.
Why must 2-ref and 3-ref systems have the same mechanic on FTs, besides some abstract notion of consistency? Refs can adapt easily enough. Besides, if you hustle, get players lined up quickly and keep the game moving, how much more time would actually be saved?
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