I suspect nobody has understood the original question, or why lean when climbing.
When the engine is operating at its most efficient operating point, which is peak EGT or just slightly lean of peak (called stochiometric combustion) each molecule of fuel is getting attached to the correct number of molecules of oxygen.
That is the point at which you want to run an engine - whether it is a car or a plane or a lawn mower.
The carburretor or the fuel injection servo (in this context they both work the same way) dispenses fuel into the engine, according to how much air is going in.
Unfortunately the system doesn't count the # of molecules of each. It would be great if it did that, and keep the ratio exactly right, but that would require mass flow measurement on the fuel, and mass flow measurement on the air.
Now, mass flow measurement on the fuel is easy, because it is liquid, and on a liquid the mass flow is the same as volume flow, and it's easy to make a liquid pump which pumps constant volume per revolution or whatever.
But mass flow measurement on air is hard, because it's a gas and expands and compresses as it feels like. It can be done, but it isn't trivial. There just isn't a "constant mass flow rate pump" for gases. The best you can do is shift the stuff as best you can while measuring its pressure and temperature and calculate the mass flow from that. That's what modern cars do, but the old aeroplane engines never went to that system, for various reasons, some good, some crap.
So, the fuel metering system (whether a carb or fuel injection) meters air by something halfway between mass flow and volume flow, and as you climb and the air gets thinner, there is an increasing error resulting in too little air being let in, relative to the fuel that's going in. So you have to lean the mixture to restore the correct ratio.
A primitive solution would be an altitude compensated carb (or fuel injection) which has a barometer and leans the fuel with increasing altitude. But this makes no allowance for temperature! At higher temps the liquid fuel is still the same "size" (avgas expands only 0.1% per degC) but the air gets "bigger" quite fast....
So, that is why you have to lean the mixture extra as you climb, to hold the engine at peak EGT.
There are some second order effects due to increasing altitude; for example the exhaust back pressure is lower so the pumping losses in the engine are lower, so for a constant MP and constant fuel intake etc, the engine will delivery very slightly more power at a higher altitude. But - unless turbocharged - normally there is such a huge loss of MP that nobody is going to notice this.
The "TAS gain" is a completely different thing. It is to do with the crude airspeed indicator progressively under-reading as the air gets thinner. But a plane pulled along by constant thrust will go faster when higher up because the air is thinner.
I wouldn't trust any performance projections from Flitestar because they come from a generic POH for the type. To use these accurately, one would need to do performance measurements. For example FS assumes 12GPH for my TB20 but in reality I get the target IAS or TAS at 11GPH, by going to LOP.
It would be possible for avgas engines to be like modern car engines, and have gas flow measurement for the air and meter the fuel accordingly to maintain peak EGT. But this would require extra sensors and complexity, and electronics in GA have a poor reliability record, and it's really easy to lean for peak EGT if one has a multi-cylinder engine monitor. You can lean for peak EGT on the ground, or at any altitude, and get the best economy that way. But there are issues with the air cooled engines: to save weight, they have thin sections and generally poor thermal design, and have to be run rich at high power settings especially when the cooling airflow is low, as it is during climb. So, even with proper fuel/air metering, one would still have to go full-rich for a max power climb.