Q.Although thermodynamically feasible, in practice, magnesium metal is not used for the reduction of alumina in the metallurgy of aluminium. Why ?

Using Mg as a reducing agent would need 3 mol of an expensive, energy-intensive metal (itself extracted only by electrolysis of fused MgCl$_2$) for every mole of Al$_2$O$_3$ reduced, plus a separate high-temperature step and downstream separation of Al from MgO. The Hall–Héroult process instead reduces molten alumina directly by electrolysis, using cheap carbon electrodes and cryolite as a flux to lower the melting point — a one-step, continuous, economical route. …

Step 1 – Note the distinction: thermodynamic feasibility vs practical/economic viability

A negative $\Delta_rG^\circ$ only tells us a reaction can proceed; it says nothing about whether it is the best industrial choice. Cost, availability of reagents, energy consumption, and ease of product separation all matter in practice.

Step 2 – Why Mg reduction is thermodynamically fine but practically poor

• Mg metal itself is far more expensive than the value of the Al it would help produce — Mg is itself obtained only by electrolysis of fused $MgCl_2$ (Dow process), so using it as a "reducing agent" simply shifts the expensive electrolysis step from Al to Mg, and then you still need to separate Al from the MgO/excess Mg byproduct.
• The reaction $3Mg + Al_2O_3 \rightarrow 3MgO + 2Al$ consumes 3 moles of Mg per mole of Al$_2$O$_3$ — a large mass of an expensive metal for a modest yield of aluminium.
• Handling a violently exothermic solid-state reduction (thermite-type) at the huge tonnages aluminium is produced at is impractical and hard to control safely.

Step 3 – Why electrolysis (Hall–Héroult) wins instead

• Purified alumina is dissolved in molten cryolite ($Na_3AlF_6$), which lowers the melting point from ~2323 K to about 1140–1200 K, and electrolysed directly using cheap carbon (graphite) electrodes. …

Separate the question of thermodynamic feasibility (sign of $\Delta G^\circ$) from the question of industrial/economic practicality — cost of the reducing agent, ene …

• Treating 'thermodynamically feasible' as equivalent to 'industrially used' — many feasible reactions are commercially non-viable. …