Intro. Based on the Gotham TV series and related media, here is how you can introduce the city without starting with Batman:
Focus on the GCPD & Crime: The show Gotham begins immediately with a gritty, noir atmosphere, focusing on Jim Gordon's first day as a detective in the corrupt Gotham City Police Department. The opening scenes highlight rising, non-superpowered crime, such as the murder of Thomas and Martha Wayne, setting up a detective story rather than a superhero story.
Emphasize a "Pre-Batman" World: The setting is a city entirely without a caped protector, focusing on established mob families (the Falcones and Maronis) and corrupt politicians. It frames Gotham as a place that is already broken before Batman arrives.
Establish the Atmosphere: Instead of a hero, the "hero" of the intro is the city itself—dark, rainy, Gothic, and dangerous, where young Bruce Wayne is merely a traumatized bystander in the beginning.