While I agree with you in general, you would have a better case with a different city than New Delhi, because most Indians simply never use the full English name in informal parlance — it's always been 'Dilli' (in Hindi) and/or 'Deli' (in inf. Indian English). In fact, the term (and equivalents of) 'Old Delhi' see more common use than 'New Delhi', largely because the latter is the one most commonly referred to as just 'Delhi'.

Talking with people who live in Delhi (in both English and Hindi), I got the sense that 'dilli' is becoming increasingly rare.

I am overall rather tired of the city-renaming craze. It seems like expensive bike-shedding at best. You end up with ridiculous scenarios like renaming 'Madras' to 'Chennai', where 'Madras' in fact is a native, historic Tamil name for the city, and 'Chennai' is the name of Telugu collaborator with the British.