when you need to raise the temperature of the mash, you can either heat the container it is in (regular method- wort pumped thru HLT), or put something hot into it (your method- hot coil in mash tun).
our objective: heat the mash to the desired temperature without going over. the [without going over] is the key here, as its easy to heat something quickly. the trick is to heat it quickly AND not go over the target temp so you dont denature proteins or extract tannins.
if you do the regular method- the temperature of the wort is the hottest at the coil directly coming out of the HLT, which is easy to measure as you can put a temp probe right here, one central location. you can control this temperature by either; adjusting the temperature of the HLT, adjusting the time the wort travels thru the HLT, or both. the larger the difference between the mash temp and the HLT temp, the faster it will heat the wort. HOWEVER- if the HLT is too hot, it will overheat the wort while its inside the HLT, which we dont want. If the HLT isnt hot enough, it will take forever to heat the mash tun to desired temperature. so there needs to be a tradeoff.
using your idea of a hot coil in the mash- the temperature of the wort contacting the coil in the mash tun would be the hottest. this is impossible to measure with any accuracy as there is no place to put a probe to measure the hottest point, so you never know if you are overheating anything. you can only evenly control the temperature by circulating or stirring the mash, constantly. if you stop stirring, the temperature stratifies and you really dont know whats going on. as mentioned, stirring is also made more difficult because you have a coil in the mash tun.
IMHO an even better method is to have a seperate heat exchanger dedicated to keeping the mash up to temp. some people use a RIMS tube, where the electric heating element contacts the wort that runs thru the tube. I use HERMS, but the heat exchanger is a small water heater with the wort coil, like a mini HLT, but since its only 1 gallon it can change the temperature much faster, and doesnt have a chance to scorch the wort like a direct RIMS heating element would.