Cooling cooked onions before adding to raw mince

Well, you're definitely right. Onions cooked at a high temperature act differently than ones cooked at a low temperature, per "the bible" aka On Food and Cooking. However, it doesn't go in to much of an explanation as to why. Probably the most relevant aspects of what is in there are:

When onions and their relatives are heated, the various sulfur compounds react with each other and with other substances to produce a range of characteristic flavor molecules. The cooking method, temperature and medium strongly affect the flavor balance. Baking, drying, and microwaving tend to generate trisulfides, the characteristic notes of overcooked cabbage. Cooking at high temperatures in fat produces more volatiles and a stronger flavor than do other techniques. (p.311)

Since a low temperature produces less volatiles, I assume that the natural sugars of the onion shine through instead.

It's probably also important to understand where the oniony flavor comes from. The spice, as you describe it, is the onions natural defense mechanism. However, it doesn't exist defacto in the onion, but is rather the result of a chemical reaction. The onion stores very reactive sulfur in the cell fluid, and a seperate trigger mechanism in a storage vacuole. When you cut / crush / cook/ peel the onion, you break the vacuole and the enzyme and sulfur cause a chemical reaction giving off the spice. (p272 & 310)

It's not so much the taste as the texture. If they haven't been sauted first, the onions stay relatively crunchy during the rest of the cooking. The same is true of the garlic, but you'd usually have cut the garlic into much smaller pieces so it doesn't take as long to soften up, hence kicking the onion off first and adding the garlic a bit later.