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It is my interpretation that injury poisons made with Quick Alchemy are potent until the beginning of your next turn, i.e. they are potent until the beginning of your next turn if not applied to a weapon. When an injury poison (normally not from Quick Alchemy) is applied to a weapon it is activated to apply its effects until (the duration) the weapon bearing poison hits and does piercing/slashing damage or until the weapon bearing the poison critically misses (at which time the poison affliction begins or does not). Thus the duration of the poison starts when it is applied (i.e. the Interact actions are taken to apply it). With Quick Alchemy, that duration (starting after the Interact action), would be 10 minutes not the open ended duration described above. So, Action #1 to use Quick Alchemy to make, let's say, Giant Scorpion Venom (which is now potent until the beginning of your next turn), Action #2/3 to apply (i.e. 2 action interact) the Giant Scorpion Venom (Infused/Quick Alchemy) to a weapon (which now will last on the weapon for 10 minutes). If you were a toxicologist, you could Action #1 Quick Alchemy the poison, Action #2 Apply the poison with interact to the weapon (which will now last for 10 minutes on the weapon), and Action #3 attack with the weapons.
Does that sound kosher?
Yes, I believe that the general consensus matches your analysis, as far as poisons created with Quick Alchemy - Create Consumable goes.
Note that this allows you to poison up weapons in advance of a fight, if you use things like Doubling Rings or a Thrower's Bandolier.