You are in such a different class than our family that I imagine your priorities, felt needs, etc are all very different so that will skew suggestions, I think. That being said, as a NYC resident who has raised a 16 year old and now 6 year old in the city, mostly Manhattan, these are my thoughts:
-First, nannies are very expensive and I don't actually believe that they are best for the growth and development of young children. A good daycare center in Manhattan will still probably set you back close to your budgeted $2,300 at this point (where do you live?). However, you will not have to deal with providing sick time, vacation time, holidays, etc to a provider or holiday bonuses, gifts, etc. Daycare centers and pre-schools can provide a very rich, stimulating, cooperative environment where babies and children can have so much more access to learning and social experiences than is possible with a single caregiver. And as the kid gets older, the costs go down. This is only a 4-5 year cost assuming you do public school so this should not be a determining factor driving you out of the city.
-What size is your current apartment? Both square footage and bedrooms. You probably don't need a second bedroom at least until the baby is 2 years old or maybe even later. The baby can sleep in a crib or co-sleeper attachment for the first 6 months-year and if your apartment has any square footage at all, it is easy to place a crib or even toddler bed in a nook somewhere with a screen. You do NOT need a lot of space-consuming toys for a baby. By comparison, our family of 4 lives in a 450sq ft apartment (5th floor walk-up). If I had a higher income, I certainly would love more space, but we are not deprived by any means. The space limitation is even less of an issue if the kid goes to daycare every day; Manhattan daycares tend to have some beautiful spaces with plenty of areas for large and small motor activity and can substitute in many ways for a large house.
-NYC is expensive because housing and daycare costs are astronomical. However, you seem to have those covered. Once those are paid, NYC living can actually be much cheaper than the suburbs. For children, especially, there is just a wealth of free activities. I love raising my kids in NYC and wouldn't have it any other way. I think they get so much more in the way of access to rich cultural resources, activities, museums, public space in which to play, etc and they grow up more independent and savvy. Just as some examples, between Riverside and Central Park there are dozens and dozens of amazing, beautiful playgrounds with massive climbing structures, unique water features, multi-level sand boxes and other unique features. Most are sheltered in tree-covered oases that are pure joy in the summer time. The Natural History Museum is donation-based - and as a NYC taxpayer you should feel no guilt about donating $1 each time you go - and is a great place to take kids. It has a discovery room with an amazing museum worker who does great things with the kids. Kids love to dance and play beneath the big whale. The dinosaur section was great when my son was 2, 3 and 4; now he's in love with the planets and volcanoes, etc. There are frequent free concerts or very low priced concerts for kids that are incredibly creative. The New Victory Theater has theater aimed at kids with tickets under $20. The Met and MOMA both have free art programs for families on weekend mornings. Wave Hill up in Riverdale is a country retreat with family art programs incorporating nature every weekend and it's free every Saturday morning. The NYPL has free story times and music groups and the children's division at the main public library on 42nd street is gorgeous with thousands and thousands of children's book - in addition to the branch library that is probably within 5 blocks of you. Mommy Poppins does a round up of events every week and lists special and unique free activities that are almost always happening. The new NYC ID will give you access to free memberships at dozens of cultural institutions if you sign up for it. A membership to the Central Park Zoo is an affordable treat and is a great place to hang out regularly with the kids. The YMCA has affordable swim lessons, sports programs and more. I could go on and on, but there is not a doubt in my mind that kids in NYC have access to so much more and that it doesn't require expensive lessons and all the other stuff that upper class NYC parents tend to pay for. Beyond food and after-school care, my budget for kids activities is probably way less than $800/year and we are out and about doing interesting stuff all the time. When you are taking advantage of all NYC has to offer, then the smaller space is much less of an issue because that is the cozy place you retreat to when you just want to relax, watch a movie and eat.
-The commute sounds like hell. Your current commute is a dream. As someone who walks from home and has her kid in school and afterschool right around the corner, the ease of it is really great. When there is open-classroom at school, it is easy to stay for half an hour and still be back at work with only 45 minutes lost. If my son forgets something or needs something in the middle of the day, it's easy to run over. Same if I want to pick him up early. Your quality of life will be so, so much higher if you live close. Also, if you have family in Connecticut then you can do the commute in reverse to go see them. A day or weekend in the country visiting family can be a great treat. A weekend with family while you stay in the city and have a couple's weekend is easily arranged. Family can come in to the city and help out with the kids - maybe not on a weeknight, but if you have an easy commute that shouldn't be as much an issue.
Obviously, I'm very in favor of the city. Of course, that's because I'm a city person. Raising my kids in a diverse area, with lots of cultural and other opportunities, having lots of time to spend in "public" and feeling part of a much bigger and less insular community are all high priorities for me. (I grew up in a small town and hated it so that is definitely part of my thinking). Schools can be tough, but there are also more options if you find your kid needs something unique in terms of support and different kinds of schooling. Your values may be different and that would definitely affect the calculus. But you can definitely afford the city - especially once you're done with daycare. (By comparison, we make about $100k combined per year and have a family of 4 in Manhattan). Also, this is obviously a personal decision, but living in the city I think planning on one kid vs multiple advantages and the city also makes that easier to do while maintaining a close social life with lots of peers/playmates.
Just my perspective.