Does India need a President!!

So finally, Pranab Mukherjee is the new President of India,
we've had 13 presidents since 26 January, 1950 so far.
But my question here is " Do we Indians really need a President?"
where they are paid exorbitant salaries and given free residence,
where expenses like telephone bills, electricity bills, transport bills and others are paid for,
where post retirement, they are entitled to pension and a government rent free bungalow.

Over Rs. 300 crores is spent during a single president's tenure,
where the same amount could be wisely spent on providing homes to the homeless,
on infrastructure or on providing proper drinking water

The president is the formal head of the Legislature, Executive and Judiciary branches of Indian democracy
but he can't summon or prorogue the houses of parliament without the advice of the Prime Minister, you see.

International treaties and agreements are negotiated and concluded on behalf of the president,
but normally negotiations are carried out by the PM and treaties are subject to approval of parliament.

Yes, the president is the commander in chief of the Indian armed forces,
but no war can be declared or peace concluded over any country without the army, navy and air force chief's consensus.

The president is the formal head of state but still seems so powerless,
it really doesn't matter if we have a president or not, nonetheless!
Cause' of late this vacancy is filled by retired ex-politicians or puppets of political parties,
Hopefully, Pranab Mukherjee does not turn out to be just a ceremonial head or a globetrotter like his predecessor,
I pray idealistically and optimistically!!