Unfortunately I think this only goes so far. You are correct that adding supply via changing zoning would definitely mitigate the effect that high-demand has on prices in the aggregate, but there are a couple issues that aren't resolved.
1) There's a major time delay here. Most rent-controlled apartments have rent increases tied to inflation. Had zoning and land development policy over the last 50 years kept up, maybe in aggregate affordability within communities could have been preserved. As it stands, we are so far behind what would have been needed to keep things affordable that it's basically unworkable in the short to medium term (10-20 years) to expect changing zoning laws to solve our problems now. NYC would need an insane capital infusion, and any economic downturn would make the building market freeze up (as we saw in 2008).
2) New development in price-controlled neighborhoods doesn't solve the issue for people paying significantly under market. That's exactly what happens now, in fact: new development once a neighborhood is rezoned allows developers to buy up old properties and build luxury high rises. There is SO much demand for luxury (i.e. Manhattan yuppie) housing that, again, in the short-medium term, long-time residents are still necessarily displaced without protection.
3) To what degree should a community be able to resist change? Other than the fear of being displaced, renters also fear their communities being torn apart by gentrification, slowly dissolved by a changing demographic and rising prices. Now, I lean more on the side that communities don't necessarily have a right to exist (i.e. NIMBYism in CA), but there's definitely some leeway here, especially when those communities are classically disadvantaged (socioeconomically, racially, etc.). Rezoning a neighborhood that is fragile can have majorly negative impacts on the communities, and make things even more dire for the most vulnerable segments of the population.
4) It's unclear how much remaining demand is unsatisfied because of housing prices in NYC. Think of the surplus of people commuting in on major rail lines each day. Surely some fraction of them live in the suburbs for lifestyle reasons, but I'm sure a large fraction of them are simply priced out of the city. So demand for housing may increase if prices are lowered substantially. Depending on this unknown demand, rezoning could have little to no effect on the populations we're discussing here.
Rezoning is an extremely important part of the solution here, and the city should DEFINITELY be making aggressive moves to rezone in various places. But it's not a silver bullet, and in the short-medium term doesn't solve most of the problems that rent control have attempted to solve.