I'm not really familiar with new york's streets, but ours have been over capacity forever and now with uber/lyft there's a lot more single passenger car traffic in the business areas. Much more so than cabs, because getting them is much more convenient than cabs ever were here - calling a cab in chicago is a joke, honestly, and if you weren't in an area they pass by frequently you were out of luck. I feel like more people choose the uber option for short trips that would have been transit in the past. The convenience of it is the model, after all.
You would need Uber/Lyft's help I believe. But I think if you combined the Uber/Lyft data with data that the DOT keeps it would be pretty easy to quantify the impact. I think that was NYC's point as well. But unfortunately the mayor had received money from a taxi lobbying effort which allowed Uber to cry foul and make accusations of protectionism.[1]
One metric used is average travel speed which according to this recent article is down 12% to 8.1 miles compared to 2010[2]
[1] http://www.nytimes.com/2012/07/18/nyregion/de-blasio-reaps-b...
[2] https://www.nytimes.com/2017/03/06/nyregion/uber-ride-hailin...
Has the city published any traffic impact studies? NYC showed concerned for Uber impact on traffic and made an attempt to get data and it turned into a political kerfuffle with accusations of protecting the taxi industry etc. I don't like the title of this blog post but it points to similar of what you are stating about downtown Chicago:
http://nyc.streetsblog.org/2017/02/27/its-settled-uber-is-ma...