The proper take-away is, I think, not to bend over backwards in trying to apply real-world logics to your fantasy game, but to ask yourself the direct question "do I and my players find poison use fun, or do we consider it disruptive to our campaign?"
If you answer "fun", then by all means introduce special containers that allow risk-free poison use, no skill check necessary. And either read the rules to mean one dose of poison applies to all hits within one minute, or perhaps that you can use your free "object interaction" to coat your weapon once per round as part of your action (similar to how monsters like Drow seem to operate).
If you answer "disruptive", then by all means use the Basic Poison as a model to restrict one poison dose to a single weapon or three pieces of ammunition, and say that once you hit, the poison is expended, and also add a skill check each time you apply poison with failure by more than 5 signaling "you accidentally poisoned yourself".
Both rules readings are equally valid. Neither is RAW, but neither is wrong either.
If the design team wanted to write unequivocal poison rules, I think they would have done so. Instead, I believe the ambiguity and incompleteness is fully intentional, so that 5th edition will appeal to both poison-lovers and poison-haters alike, without scaring either group away.