The actual statistic is 70% of households don't have access to toilets and 60% defecate out in the open. 60% of 1.2 billion people is quite the amount of feces getting around.
I've been to India twice for hair transplants, and I did see some public defecation, but only when straying from the city. I saw dozens while in a car going through the countryside, but not while walking around.
Remember that India is a big country, so the percentages aren't even across all of it. If OP is thinking of visiting someone like Dr. Parihar, the clinic is in Mumbai. Mumbai is the most populated city in India, and is considerably more metropolitan depending on the area, which means better access to restrooms.
Back on topic, auto rickshaws are everywhere and they're cheap; just make sure that the meter is running (illegal to operate if broken), and don't let them suggest destinations for you (cheap jewelry, massage, cute girls, etc... Driver gets a cut). It only happened once, but a driver kept suggesting a different hotel to me when I specifically asked him to take me to another; be strong, don't give in, say that you have a reservation. Even hiring a driver isn't that expensive. And yes, Uber is a thing over there if you're used to ride share services back home. If you're ever concerned about being ripped off, ask the hotel concierge to find a driver for you.