Firstly, it would be better to use actual, accurately measured numbers than the human 'experience': the human body is a poor thermometer and the mind plays tricks on us.
But that hot water droplets lose heat and thus cool down in cooler air is an established fact and a consequence of the laws of thermodynamics.
Regarding your three first bullet points, despite some limitations you point out, those do not mean a hot droplet of water doesn't cool in air: it does and partly in accordance Newton's cooling law.
As regards work done by the droplet (overcoming the viscous drag), if anything that would lead to heat generation, not cooling (but the effect is truly minuscule).
Kinetic or potential energy of the droplet have no effect on the droplet's temperature. Temperature is simply a measure of the average speed of the molecules of the water and that is not affected by these energies. Spinning water in an ultra-centrifuge does not make its temperature rise, for instance.
You have however overlooked one major cause of heat loss: evaporation. Your shower 'steams up' because hot water evaporates and that costs energy, known as the Enthalpy of vaporisation.
Millions of tons of water are cooled this way everyday in power plants world wide: the cooling towers drop hottish water from the top of the towers and evaporative heat cools down the water (the evaporated water escapes as steam clouds).
If your shower has been in operation for a long time and the bathroom's temperature is equal to the shower head's water temperature and the air is saturated with water vapour, then no cooling of the shower water would take place.
I will assume your heat exchanger uses the common counter flow principle - that is, the direction of flow of the "cold" water opposes that of the "hot" water, so the hottest water in the heating loop (entering the heat exchanger) is in touch with the hottest water of the pool loop (just before exiting the heat exchanger).
The heat flow across the exchanger is proportional to the temperature difference. Since the "input temperature" is fixed at 180 F, the only variable is the temperature of the "sink" - the pool water. The colder the pool water, the greater the heat flow.
At the input of the heat exchanger, the temperature is the temperature of the pool; at the exit, it will be somewhat warmer. The slower the water flows, the more heat it will pick up, and the hotter the water that re-enters the pool. However, the hotter the pool water in the exchanger, the smaller the thermal gradient, and therefore the smaller the heat flux into the pool water.
The water will heat most rapidly if the pool water runs quickly - this keeps the temperature difference greatest.
There is just one caveat: the power of the pump moving the water. If the pump is working harder to move water through a constricted valve, it would generate a little bit more power; if the water flow is set up so heat from the pump is dumped to the water, you will get a small amount of additional heating; but I don't believe that would ever offset the benefit of the faster gradient.
One other consideration: what happens to the surface of your pool. This relates to the way the output of your heat exchanges returns to the pool. If you have a jet that dumps deep inside the pool, there will be little disturbance at the surface; if it's aimed at the surface, you will cause some "stirring". As you may know, the greatest heat loss from a pool happens through evaporation - so if there is anything in your setup that increases evaporation as a function of flow rate through the heat exchanger, that will affect the total heating time.
If it were my pool, I would probably rig up a thermocouple and a data logger, and look at the evolution of temperature. Turn the flow rate up and down every two hours or so, and see if you can observe a change in heating rate on the temperature trace.
I am sorry - according to the laws of physics, your pool guy is wrong. Open that valve, and let the water flow!
Best Answer
I think you understanding of how the hot water heater (not a boiler, incidentally) system works is way off, if you don't mind my saying.
The temperature setting means that the heater will kick on until all the water in the tank is at that temperature, then it switches off. Just like your house thermostat. When you use the hot water and drain the tank below a certain level, the tank starts to replenish itself using cold tap water that needs to be heated up. So you end up with a mixture of gradually cooler and cooler water as you keep running the faucet.
So yes, starting with hotter water will help it stay warmer for longer. But whether this makes a noticeable difference or not depends on the sizing of the tank, the cold water temp, the heater power, and other things. And I would not recommend messing with your heater setting too much, as it could shorten the life of your system. Better to get a bigger tank or simply allow more time between uses.