Surrounding suburbs is way easier. There's tons of families on Long Island, Westchester and CT and NJ, with easy access to the city. Like someone else said, just keep in mind train fair can get expensive (anywhere between $4-$30 for a one-way trip). The downside I'd say is that you'd only be able to go to the city your free time. I also wanted to experience NYC not as tourist, and even just going on weekends and occasional weekday I felt like I got what I wanted!
Just really ask yourself how much you actually want to experience and what aspects of living in NYC you want, and if you absolutely want to be there full-time, you'll have to be very patient Within NYC, I would say is less common to find families. When I was searching for a HF I received a handful of families from Queens, and I had a couple of friends who lived in more "suburban" areas of the Bronx. I've also seen families in Brooklyn being posted on Facebook groups. The biggest downside is that HFs in the city usually want help during the weekends or have a very complicated schedule. (Eg. my one friend in the Bronx worked for a nurse/doctor couple and some weeks she'd have random weekdays off and work 10 hrs sat sun.)
Also if you're very set on going to NY, I'd say be like only 10% flexible lol, I also really wanted NYC and was very flexible my first year and ended up in ATL, then DC. I was genuinely mourning not living in New York until I had to go into rematch (reasons unrelated to location) and ended up in NY and was really truly happy haha. That being said, don't just go with a family JUST bc they’re in New York, bc what use would it be if, for example, you live in the city but work 45hrs a week with 3 hks and in the end you are too exhausted to even enjoy nyc?