I'm not nearly as familiar with that area as you are. But a little digging suggests there is much more to the picture. Nassau is notoriously much more restrictive on zoning and, in fact, zoned almost exclusively for single family homes, which means that the rentals that do exist are going to be of entire homes rather than apartments. Queens is only 25% zoned to single family homes where it is more like 95% in Nassau County. So far less SUPPLY of apartment rentals in Nassau because there are less apartments, and those rentals that exist are going to be larger homes not apartments. Apply the same zoning between Queens and Nassau and you'd probably see a leveling out of the difference over time. But even so, just looking at rent calculators it looks like rents are lower in Nassau than Queens. There are other issues such as different school districts that may make renting on the Nassau side of the line more attractive.
People study this subject in depth. And what do they find? That there is an almost perfect inverse relationship between vacancy rates and rents in most housing markets in the country. For example, here is Austin TX. During the 2008 recession when the tech sector suffered and vacancy rates went up in Austin, rents immediately dropped. When Austin became the hot tech market again in the early 2010s, boom, vacancy rates went down and rents went back up. Taxes were unchanged during this entire period.
You can do this for any city in the country. Here is Denver
Bottom line? Want to reduce rents (or rent increases) in an area? Allow more housing to be built. Supply and demand. Due to inflation, rents do tend to go up over time. No one charges the rents from 1970 or 1950. So the real issue for housing affordability is keeping rent increases at or below the rate of inflation.
Likewise, if you want to reduce the price of homes in an area (or prevent big increases) it it also about supply and demand. Although that is a bit more complex because you have lots of factors affecting supply of homes for sale beyond just the quantity of existing homes. For example, right now a lot of people who might otherwise sell are locked into their homes by rising interest rates. They might have a 2.5% mortgage on their existing home and don't want to sell because any new home they buy will have more like a 7% mortgage. So the rental market is a bit simpler.