Temperature is the average kinetic energy of all the molecules in a system.
Thus \begin{align} T &= \frac{1}{N}\sum \frac{1}{2} m v^2 \\[6pt] &= \sum \frac{E_\text{single particle}}{n} \cdot 6.022 \times 10^{23} \, . \end{align} where $N$ is the number of particles and $n$ is the amount of substance.
The unit here is J/mol. Why isn't this used in place of temperature? (Instead of K, °C or °F)?
Temperature is the average kinetic energy of all the molecules in a system.
It’s not that simple. Let’s first look at an ideal, mono-atomic gas. Here we have:
$$ T = \frac{2}{3 k_\text{B}} \bar{E}_\text{kin},$$
where $k_\text{B}$ is the Boltzmann constant and $\bar{E}_\text{kin}$ shall be the average kinetic energy of each particle. Apart from the proportionality constants, this is what you were describing. However, the $3$ in that equation is the number of the degrees of freedom of each particle. If we consider an ideal gas whose particles are molecules composed of two atoms (e.g., oxygen), we have five degrees of freedom and thus:
$$ T = \frac{2}{5 k_\text{B}} \bar{E}_\text{kin},$$
In real situations (i.e., with non-ideal gases, fluids, and solid states), things become much more complicated here, as there are various degrees of freedom with differing properties and all of this becomes temperature-dependent on top. This is what is usually expressed in the heat capacity of a material.
Therefore there is no simple way to define temperature using kinetic energy. Now, you may ask: “Why is temperature defined such that we have to bother about heat capacities in the first place? Couldn’t we just define it via the average kinetic energy despite all of this?” In theory, we could of course do this, but then we would lose one of the most practical properties of temperature, namely that two bodies with the same temperature are in thermal equilibrium. For example, two materials which melt under the same conditions would very likely have a different melting "temperature"¹, or every object in your room would have a vastly different "temperature". Also, it would be very tedious to measure that "temperature".
¹ in the new sense
If you read this report from the body who are responsible for the definitions of SI units (BIPM) you will realise that yours is an excellent suggestion made a few years too late.
The only difference is that BIPM have not used the mole in their definition of the kelvin:
The kelvin is the thermodynamic temperature at which the mean translational kinetic energy of atoms in an ideal gas at equilibrium is exactly $(\frac 3 2) \times 1.380 65{\rm X X} \times 10^{−23}$ joule
or variations of that statement, for example:
The kelvin is the change of thermodynamic temperature that results in a change of thermal energy kT by exactly $1.380 65{\rm X X} \times 10^{−23}$ joule.
or
The kelvin, unit of thermodynamic temperature, is such that the Boltzmann constant is exactly $1.380 65{\rm X X} \times 10^{−23}$ joule per kelvin
With, I think, the last definition favoured at the moment and reasons for not adopting the other definitions are given in the BIPM paper.
So the plan is to measure the Boltzmann constant as accurately as possible and then state that that value is fixed (cannot be measured just like the seed of light in the definition of the metre).
That is why the "X X" appears above as the defined value has yet to be fixed.
Again as for the newish definitions of the metre and the second it will mean that current temperature measuring devices will not have to be recalibrated.
The reasons for this new definition are given in this sentence:
The definition will be generalised, making it independent of any material substance, technique of realization, and temperature or temperature range, to ensure the long-term stability of the unit.
