Last thousand years? The Ghaznavid Empire was headquartered in Lahore over 150 years before it reached Delhi.
Despite the other empires that were headquartered in Delhi, were they the forefathers of modern Indians or foreign Muslim empires?
The Europeans, Arabs and Chinese used Indie, Al-Hind and Hintoi to refer to the Indus, before the term was artificially expanded.
Even Shashi Taroor claimed Pakhtun pilgrims in Arabia were referred to as "hindi" people which still translates to Indus people and not the people of Delhi.
Delhi is not the region of the Indus. Never was.
Just because foreigners artificially expanded the term later on, does not mean the natives of the Ganges and the rest of the regions called themselves such or their lands such.
British already offered the Muslim league the title India and they rejected it on the grounds that it's a foreign name. And no Pakistanis don't reject it.
All those whom are aware of it accept it. India is not an ambiguous term. Much like Macedonia.