This page on Angels' Cup has (see the middle of the page) a concise summary of the key variables and results in brewing coffee with a given method:
- If it tastes sour: The problem is under extraction, so adjust brew time longer, or adjust water temp cooler, or adjust grind size finer.
- If it tastes bitter: The problem is over extraction, so adjust brew time shorter, or adjust water temp hotter, or adjust grind size coarser.
- Temperature affects the sour notes significantly more than the bitter notes.
… you have to be willing to experiment. Don’t think of it as a waste of beans. Gulping coffee without paying attention to flavor is a waste, experimentation is the best possible use of good beans.
I've tried varying the grind size and water temperature in an AeroPress, and so far my results agree with this.
In particular, I've found that brewing medium roast beans with 175°F water (using a calibrated thermometer) produces coffee that's not sour, while higher temperature water often produces sour coffee (tasting a bit like vinegar). The company recommends 175°F except with light roast beans, in which case some people prefer 185°F water. See their FAQ about temperature.
I have not tried varying the brew time. The inverted method lets you increase the brew time but you can accomplish that without the mess by inserting the plunger, pulling back slightly to decrease the pressure below room pressure, then waiting.
You can see that the brewing variables are interdependent. That probably explains why people using the inverted method to brew longer would use hotter water to compensate for what would otherwise be over extraction. (I don't know why cooler water extracts more than hotter water.)
The ideal brew time, temperature, and grind size will vary with brew method. AeroPress immerses the grinds in water, thus it's relatively insensitive to grind size.