I always was taught that as a sling psychrometer was spun, water would keep evaporating by taking heat from the thermometer, and it would keep doing so until the air became saturated and could no longer "take" any more water.

But that didn't add up right. Because a sling psychrometer is moving, bringing in new air continually.

I guess I just handwaved it away as "maybe the air envelope around the psychrometer all became saturated"?

But something about that still didn't jive with reality... and maybe more importantly, I would think as long as any extra amount of air can be brought in, the thermometer would keep dropping below the true wet bulb value. You've got an unlimited supply of air to saturate.

But my issue and yours really go back to how the temperature stabilizes.

The fourth answer to this Quora question, by Ashish Kori, as well as this video from a professor at Lund University in Sweden, are what helped open my eyes better.

Basically you're starting with a thermometer covered with a wick that has an excess supply of water, and you "continually expose" it [i.e. by fanning/moving it] to an excess supply of air. And the air, thermometer, and water all start at the same temperature.

The movement encourages evaporation. The evaporating water takes energy from the water... so the water cools.

But as that process continues... the cooler water is now also being continually introduced to air that is warmer than it, so there is a steady transfer of energy back into the water.

The movement and unlimited water and air means the process continues until the energy transfer is balanced. The drier the air, the greater the rate of evaporation, the larger the temperature drop before the energy balances.

Why does all that matter? Because the thermometer is only a bystander in the energy exchange. The water is what is getting to the wet bulb temperature.

Sure the thermometer (both its fluid and casing) will be factors in the energy balance for a time as it does have energy to add to the equation by being briefly warmer than the cooling water... but with the unending evaporation, that bit of extra energy is very quickly forgotten, regardless of how much of it was stored (i.e. the thermometer's heat capacity).
A huge heat capacity might mean in theory that it takes slightly longer to reach equilibrium by overcoming the extra energy, but even then it really shouldn't be significant. (I would think even the thermal conductivity wouldn't be a big factor in a sling psychrometer because of all the mixing motion, so long as the wick isn't so thick as to significantly insulate the thermometer.)

Aspirating ensures that any energy buildup (heat capacity storage\conductivity delays) in both dry and wetbulb thermometers is overcome.

The thermometer is only a bystander in the evaporation, and it must eventually reach the temperature that the water reaches. The thermometer's heat capacity is unimportant to the continuing energy balance.