Nearly all of the Westchester suburbs are extremely walkable. All of the essential stuff is on the train line. The wife and I would load up the baby and take the train to Portchester, where there was a Costco, Marshalls, etc. I did have a car, which I drove maybe once a month (the upscale mall in White Plains was nowhere near public transit).
A 35 minute commute is not bad at all. Lots of rich people all over the country have hour+ commutes from the suburbs. Heck, my commute from New Rochelle was a lot more enjoyable (straight shot into GCT) than my commute from the UWS (transfer at Times Square, ugh). You can't point to 35 minute commutes as a reason to make space for low-income housing in the city proper.
I mean, I totally get what you're saying, but I didn't find much benefit to living out there. I don't have a family though.
In any case, you're arguing a different point now. Housing prices in many Westchester towns are very reasonable and commutes are too. No reason for rent control to allow low-income people to live in the city proper.
Now granted, their commute is going to be longer since that 15-30 minutes is in addition to the rest of their commute, not the entirety of it, but you can't live outside the city you are working in and expect snappy commutes.
Also, can you really live in these areas and shop without a car? I find the answer to that question to rarely be yes.
I shop with my car maybe once a month, tops, and that is mostly just because I like to buy dozens of liters of seltzer water.
How difficult is it to live where you do without a car?
I was mostly speaking about the areas near the train station that you mentioned yeah. Those prices are pretty high for the commute time that you end up having. A 15 minute walk is okay but any longer than that might be very difficult for some people.