Now what is a little confusing is the part about not getting outside oxygen into the carboy, yet aeration is in essence oxygen.
What's confusing? Air is ~21% oxygen. If you aerate (i.e. dissolve air), you're getting oxygen as well as other gases (mostly nitrogen). The only one (gas) that yeast need,
if they are currently in a state that needs it, is oxygen. You can oxygenate the wort more efficiently, i.e. faster and more thoroughly with pure oxygen rather than atmospheric air. Once the yeast are in a state where no O2 is required, any additional O2 is free to oxidize (stale) the wort. It's the reason why O2 is generally only added at (wet) yeast pitch time (when the yeast can fairly rapidly assimilate it), and not later.
My school of thought is that the more aeration you have AFTER pitching your yeast then the better the flavor outcome.
Oxygen in excess of yeasts' needs does not improve flavor, unless you want the taste of staling compounds. Some styles, such as big stouts, can benefit (depending on preferences) from certain (but not all) staling compounds. "Fortunately," aging tends to produce those, because it's impossible to
completely exclude O2 later.
To surmise you just want your yeast to have plenty of sugars to munch on, and oxygen. Aeration for a minute or two after pitching the dry yeast is advisable.
Yes, yeast need sugars to survive (and to make beer), and they need oxygen (or a direct source of sterols)
sometimes. They also need lots of other nutrients, such as "nitrogen" (in the form of amino acids or other compounds (not dissolved gaseous nitrogen), zinc, and others).
But aeration for "a minute or two" (or any amount of time) after pitching dry yeast has not, to my knowledge, been shown to be advisable, and in fact is not generally advised.