Chemistry - If LPG gas burners can reach temperatures above 1700 °C, then how do HCA and PAH not develop in extreme amounts during cooking?
It is the temperature of the pan that matters not the flame
The flame temperature is irrelevant if you are cooking in a vessel. the only temperature you need to worry about is the temperature of the surface of the pan (or–even more importantly–the temperature of the meat). The surface of the pan will rarely get above around 220 °C if you are monitoring it. An unattended pan can get hotter, but, if you let it get too hot, your food will rapidly burn and will not be edible. Conversely, if the food is palatable, you probably haven't heated it enough to get lots of nasty HCAs and PAHs.
But the research that caused your worry is also frequently overstated. Yes, large quantities of PAHs or HCAs may be nasty, but the amounts in food–even food cooked on an open flame barbecue–are very small and there have been no convincing studies showing a notable effect on health. There are some studies showing a very small effect of meat on health (diseases such as bowel cancer have been linked to some meats but the studies are statistically weak and the effects are very small. Moreover these studies link to meat not cooking products but would probably have spotted any effects based on nasty cooking by-products). It is worth remembering that people have been cooking on open flames since we invented fire. If that were really dangerous, there would be strong evidence of harm and/or primitive man would have developed good defences to avoid the harm (as milk-drinkers evolved lactose tolerance in adults because their diets consisted of a lot of dairy products).
There have been many food scares based on observations of known nasties in cooked food. Acrolein and Nitites, for example. None of these have been shown to have any notable measurable effect on people in the concentrations present in the diet.
Nothing with more thermal mass than a needle will reach the temperature of the flame. Note that if the stove or utensils would be this hot they would be glowing blinding white, while IRL they won't even glow a dim red. Granted if you dripped fat on a hot stove it would turn into a black tar containing PAH's, and a microscropic fraction of this could end up in your food. We are exposed to carcinogens and radiation all the time, theres no reason to be alarmed.