The problem with the Gale-Shapley algorithm for stable matching is that it's inherently unfair. The askers/proposers slowly get downgraded to their lowest possible match, while the other side slowly moves up to their highest possible match. Not sure how different the outcome would be from the current system.
Which actually brings me to another point: The GS algorithm is roughly that members of one party(in this case the schools) work down their list of preferences, asking members of the other party(the students) to partner with them. The members of the other party always say yes to their first match, and will always leave their current match when someone higher on their list asks them to be partners. Sounds pretty similar to the system currently in place in India.
A final note on the GS algorithm: If an anecdote told by a professor of mine is correct, this has been tried before. It was how hospitals chose their residents after med school, but it was set up so the hospitals always got the best match. I believe the students sued them.