Would a thermometer calibrated for water also be accurate for measuring the air temperature (or vice versa)?

Temperature is a measurement of the average kinetic energy of the substance -- so if we have a thermometer calibrated for water, could it be used for measuring the temperature of air? My inclination is to think no, because the amount of kinetic energy that is being pulled from the thermometer (e.g. 1 degree C water) is going to be much greater in water than in air, but I'm not sure if I am missing something.

Yes. At equilibrium everything has the same temperature. Particles on average have the same kinetic energy. That is, particles with above average kinetic energy collide with particles that are on average less energetic. The more energetic particles tend to lose energy and less energetic particles tend to gain. This works even if different kinds of particles are mixed together.