While boiling eggs this morning for my brunch, I remembered that my chemistry teacher in High School used to say that the tip of the flame was the hottest part of the flame. And so, on a gas stove, you should only turn the heat up to a level where the flame just touches the bottom of the pan (for maximum results.)
The tip of a gas flame is the hottest part, so he got that right.
There are a limited number of flame jets (6 or 8?) in a gas stove burner. If the flame is turned up so the flame just touches the bottom of the pan those 6 to 8 points will be heated by the highest temperature, but the majority of the pan will be heated very little.
By turning up the flame more, the average temperature of the flame may be lower than the flame tips, but there will be a lot more heat transferred. In the same way, even though a match is a lot hotter than the air put out by my heater, I wouldn’y try to heat my house by lighting matches.
I think that rule only applies to orange (or red or yellow) flames. The orange part has insufficient oxygen for complete combustion, resulting in a lower temperature and the orange color. At the tip of the flame the “fuel” and oxygen are mixed thoroughly and there is complete combustion resulting in a higher temperature. On a gas stove, the flame is fed with a mixture of gas and oxygen so there is complete combustion in all parts of the flame.
That’s one of the main differenced between pre-mixed flames and diffusion flames.
In burners like oxy-acetylene torches the fuel and oxidizer are blended in proper proportion before ignition and are limited only by propagation rate. That inner core is the limit of the flame wall where the flow rate of the fuel/oxidizer mix perfectly balances the propagation rate of the flame, and that’s where the real action happens. Same thing with Bunsen burners, but the oxidizer is atmospheric oxygen.
Diffusion flames, like candles or the jets on your stove, are different. There you have hot fuel on one side of the flame wall and oxidizer (usually atmospheric oxigen in most applications) on the other. They interface at the flame front and diffuse together to react, so the tip is in fact the hottest point.
As far as the OP goes, you don’t care about flame temperature as much as total heat generation. While the tip is the hottest part, just getting the tips in contact with the bottom of your soup pot isn’t going to transfer as much heat as a larger volume of hot gas.
More fuel burned gives you more heat.
This answer is really general and kind of lousy.
Someone like Anthracite could give you chapter and verse.