First paragraph) yes vision is incredibly valuable, but not when you know that their red side jungle for instance is preoccupied taking their buffs for the fist 2-3 minutes of the game. Between you and your adc, you have two yellow trinkets. That's two minutes worth of vision that collectively you can work with, and Imo that is enough to protect you until the first back.
What is your definition of "proper mana management"? As far as I'm concerned, it involves taking into account anything to do with mana I.e mana regen items, mana cost on spells e.t.c. And how and when I use it. That includes mana pots.
I disagree entirely with your conclusions in your second paragraph. There are a lot of factors that you haven't taken into account. if you have twenty five percent health and no mana, and they have a support with 75 percent health and no mana, but then they use a mana pot, then who is "winning" that 1v1? I don't like to use winning really, but we could probably agree that in that instance the enemy support has dominance over that lane.
What do you mean by "negated with my mana"? It depends on the matchup. In some match ups, janna shield can be broken with ease plus additional damage, for example zyra, so overall I don't really know what your point is with paragraph two.
Third paragraph) No. What I was trying to say with my original example is that there are situations where you will need to use mana (for whatever reason that may be) when you don't expect to or ideally wouldn't want to, and because you don't take a mana pot, you can be put at an unnecessary disadvantage. I was trying to justify this by saying that the ward you bought instead would be suboptimal because it provides vision in a place where you cannot contest if you see the enemy e.g. Start dragon, invade through tribush, since you have less mana than would be required.
I cannot confirm or deny statistics about " experienced players " and to be honest neither can you. I think I could find somewhere a pro ( if that counts as experienced) guide that has a starting build that includes one mana pot.