Heat rises, but the bottom is closer to the heating element. Where would the hottest point typically be?
Bottom. The boundary conditions would require it. If the heating element wasn’t supplying enough heat to keep it that way, then the water would stop boiling.
I don’t have a thermometer and pot of water handy but I’ll speculate for a second. As you said, the heated water will rise. As it rises, it is rubbing elbows with water that is cooler, and will lose some heat to conduction on the way up. Once it reaches the top, it will lose more heat to evaporation on the top. So the water will be cooler by the time it reaches the top than it was on the bottom when it was close the the heat source.
So the hottest point will be the thin layer of water at the bottom of the pot.
Another, very small, effect is the pressure. The water at the bottom of the pot is at a slightly higher pressure (due to the weight of the water above). This will raise the boiling point of the water slightly. For any reasonable height pot, this effect is probably immeasureable.
You can see this if you closely watch a just boiling pot of water. The steam bubbles form at the bottom and get larger as they raise to the surface.
So, steam bubbles getting larger means that water is evaporating. As water evaporates, it requires heat for the change, so the surrounding water must be losing some heat.
No additional water is evaporating. The bubble is getting larger because the pressure is decreasing, so the gas (water vapor) is expanding. This will cool the gas (again very slightly). The effect on the surrounding water would be even smaller.
So, in that case all the steam forms at the botton (heated) surface, then rises to the top as bubbles which increase in size. Presumably the main reason which the top layer of water gets cooler is loss of heat to the atmosphere, not heat loss to evaporating steam.
I have not seen a detailed thermodynamic analysis. But, I believe both effects are significant in cooling the top surface.