That's not living somewhere. That's visiting.
If that's not living, then where do I live? I don't have a long-term lease or anything similar. Do I not deserve the same quality of life and opportunities as people who happened to have been born in the East Village?
No, no more than I deserve a passport from every country I have ever traveled to.
Most US states consider you a resident if you stay more than 90 days. That's when you'd be expected to change your driver's license, car registration, etc.
You seem to be conflating the ideas of birthright and residency. I don't think anybody is arguing you shouldn't be allowed to become an NYC resident if you want to; what they're saying is that visiting doesn't make you a resident.