What is the temperature of evaporating water (not boiling)? The vapor pressure of water at $\pu{70 ^\circ C}$ is $\pu{233.7torr}$.
If I were to put a temperature probe directly above the liquid water, what would it read?

In other words, does a water molecule need to have $\frac32kT$ (where $T=\pu{373 K}=\pu{100 ^\circ C}$) to escape the liquid?
No, the average thermal energy doesn't need to be the one corresponding to 100C.
Solo water molecules can exist in vacuum or in a gas phase with whatever kinetic energy. Liquid water condensation starts when the concentration of the water molecules is enough high (there is a tipping point), i.e. when the partial pressure of water reaches the vapor pressure for the given temperature.

From the classical thermodynamic point of view (similar argument to Hexacoordinate-C's comment), there is a chemical potential of water in the gas phase which is dependent on the partial pressure of water, and there is a chemical potential of condensed water for a given temperature. When the partial pressure is low, the gas phase chemical potential is lower than it would in the condensed phase -> it is stable in gas phase. When the partial pressure reaches the vapor pressure, the chemical potential is equal to the gas phase and condensed phase.