The perception of "spicy" is completely dependent on the person. And there is a "tolerance building" thing going on too: what is intolerably hot for somebody not accustomed to heat might not even register for somebody who eats hot frequently. As an anecdote, I had a friend who started eating hot. After a month, she started asking the cook to use pure tabasco instead of tomato sauce on the pizza. After another month, she complained that the pizza was made with tomato sauce and the poor cook had to let her taste her from the bottle to convince her that the pizza was indeed made with pure tabasco, it just didn't have the same "kick" for her after habituation.
From your description of your soup only feeling hot enough for you after adding six habaneros with the seeds, it seems that you have developed an extremely high heat tolerance. And this would be an explanation for what you observed.
Six habaneros will contain a certain amount X of capsaicin. When your soup is fresh, the pepper has enough capsaicin concentration that you notice it and enjoy it.
But when you leave it alone, the broth dissolves the capsaicine (unlike pure water, the fat and maybe also the solvents from the vinegar enable it to do so). So you end up with a kind of "capsaicine tea".
But remember that you have the same amount X of capsaicine, this time diluted in litres of soup, as opposed to sticking all in one bite of pepper. So, now that you bite on the pepper, you don't get the heat. But when drinking the broth, its concentration is so low that it does not register for somebody with a high heat tolerance. From your subjective point of view, the heat has "disappeared".
The original intention of the recipe author was certainly that the whole soup becomes somewhat hot. And for people with less heat tolerance, it will work well. They will be able to eat the broth and notice the heat, even with much lower quantities of hot pepper.
For you, it seems that you still haven't reached the dose which will work for you in a soup. So you can add more capsaicin - either use more peppers, or find a source for hotter ones. If the logistics are a problem, buy them dried. But note that at no point will the whole soup taste as hot as the single fresh pepper (due to the dilution), and also that you will become habituated with time, just like my friend with the tabasco pizza.
Best Answer
There's nothing you can put into the food that will neutralize the spice after you eat it. The only way to avoid the infamous "ring of fire" is to add less spice to begin with. One way you can do this is by removing the seeds. Some of the heat from chilis comes from the seeds, however it doesn't all get extracted from the cooking. After you eat the chili the seeds keep producing capsaicin, meaning your insides get more spice than your mouth. If you de-seed the chilis you need to use more as you lose some of the heat, but you'll get the full force when you eat the food. You can also use chili powder for most of the heat, and then whole or chopped chilis for decoration and texture as the capsaicins from the powder will be more fully extracted by the cooking process.
Some people drink yogurt based drinks (mango lassis and such) to counteract chili. This can calm the heat in your mouth and some people get some relief from the internal symptoms as well, however it's different for everyone and YMMV.
Still, there's only so much you can do, if you like really spicy food you have to be willing to pay the piper.