A lot of the Kindergarten application process is explained in Getting into NYC Kindergarten by Alina Adams. She goes into the district/zone system, as well as how private school applications work. There's also info about G&T, etc., some of which has to do with the history of the program as a tool for preventing white flight to the suburbs.
As someone else noted, there's a zone/catchment system for to get you higher up in the list of getting into your neighborhood school. It's not guaranteed (e.g., sibling placement will be higher priority than your child if you've just gotten into the zone), but most people will get into their zoned school, even if you wind up on the wait list (a lot of people will be on the wait list, but have also placed into private schools, etc.)
Note that this zone system is for K-5 (excluding G&T stuff, which is another ballgame, and may be going away anyway). Middle school stuff is different, as you can apply to different middle schools, and again for high school.
We also have the mifold, and it has been a lifesaver. Although we have a car, I use it whenever we’re not driving, when we travel, etc. The smaller one is pretty light and you can stick it in a tote bag without taking up too much space. I’ve never seen the highback one but it’s so cool!
I’ve also tried this one (BubbleBum Inflatable Booster Car Seat | Patented Stability System | Narrow, Backless, Blow up Booster Seat for Kids | Comfortable, Compact & Uber Convenient for Everyday Use & Great for Travel | Black https://www.amazon.com/dp/B00AQYZCXK/ref=cm_sw_r_cp_api_glt_i_W7J42HVQS5Q2GTF1HHST?_encoding=UTF8&psc=1). Lighter and in some ways easier.
I’ve also seen this one (Ride Safer Travel Vest Gen 5, Large, Black https://www.amazon.com/dp/B07WHV7W7F/ref=cm_sw_r_cp_api_glt_i_3W5G4KCNCF00KKJ6XRG7?_encoding=UTF8&psc=1) which looks super cool but pricier (and maybe bulkier). Depending on the age of your child (sounds like he’s probably pretty young if he’s just outgrowing the infant seat), I would only go for this one since it’s a harness rather than a booster.
I will say that I didn’t feel comfortable using the mifold and bubblebum with my kid until he was probably at least 6 because they don’t have a five point harness…even older for anything more than short trips because he had a hard time with sitting safely without a harness which made the trip stressful for me, him, and probably the driver lol…not a great combo.
Also, if you take it on a trip with you by plane, you gotta take it out of your carryon when going through the bag scanner. It will flag you each and every time, and I’ve never had someone know what it was without explanation or demonstration.
Starting age 5, we used the portable carseat Mifold for quick trips. Car service with car seats were unreliable at the time but at least drivers were patient enough to allow me to get the carseat setup.
I just looked it up on amazon but also noticed they had this option which might serve you better if your child is younger. https://www.amazon.com/hifold-fold-highback-booster-Racing/dp/B07X7H9HL3/ref=sr\_1\_7?crid=9CVNNJBSAXS&keywords=portable+car+seat&qid=1642773194&sprefix=portable+carseat%2Caps%2C83&sr=8-7
I think renting gives you much more flexibility, because you're not tied down to one place. This may be important for schools in a few years.
So our story: we moved to an apartment in the PS 199 zone on the UWS. A year after we moved, our building got rezoned to a more distant, possibly crappier school. We had the freedom to move (and, also, our property values weren't slammed for multiple tens of thousands of dollars because of the zoning change), and did to get into walking distance of the private school that seemed like a good match. The private school turned out to be a poor match because of special education needs. We haven't moved to be near the special ed school (partially because of COVID, partially because the subway commute isn't that bad at all), but we can if we need to.
One lesson we learned is not to be particularly tied to any particular school plan for kids that run many years into the future. The school you first find may not be a great fit (e.g., progressive vs. traditional education). You may wind up needing special education services. There are a lot of reasons you will want some flexibility in location.
There's a book from Alina Adams which talks about the various things going in with NYC primary school entry. Take a look at it with some notion of the value of flexibility.