Why isn't Mg deposited on the cathode after electrolysis?
You are in aqueous solution. Everywhere where there is an aqueous solution, you have a certain concentration of a further cation: $\ce{H+}$. It has a potential of $\pu{\pm0 V}$ by definition, thereby being nobler than $\ce{Mg^2+}$.

Thus, hydrogen gas is ‘deposited’ on the cathode first and bubbles away while the solution gets removed.

It is not possible to deposit considerably less noble metals on the cathode in normal, aqueous solutions.