First - take a step back and breathe. I feel your pain - I've been there. The good news is a few years down the road I can tell you my DD is now thriving. It can take time, patience, thinking outside the box and determination but your son CAN get there.
My DD has 11 diagnoses - every possible LD and processing issue. Public school was a nightmare for her - anxiety off the charts. Even with a supportive sped teacher she didn't make progress and regular classroom teachers were just in over their heads. Add in a horrible principal who didn't believe any of her challenges were real and intentionally antagonized her anxiety and it was truly awful.
We went the route of Out Of District Placement at a special ed school. No the peers weren't really appropriate but she could get solid remediation for her LD issues and they were used to dealing with anxiety. Most importantly they embraced radical exceleration in her area of biggest strength. (They placed her in an 8th grade English class as a third grader and started a HS curriculum 1-1 as a 4th grader. All using audio books.) It was like pulling teeth to even get her to read at a kindergarten level. She had a fabulous reading intervention teacher though and made HUGE progress once we got really good vision therapy. By 5th grade she was reading at grade level, by the end of 6th grade she was reading at an adult level. It. Can. Be. Done.
Once she outgrew what the special ed school could do for her we faced a difficult decision of where to place her. PublIc school was way too chaotic for her processing issues, no special ed school could meet her high level needs. Regular privates couldn't meet the LD needs. I found a few interesting things that could potentially be helpful for you.
Eagle Hill in Greenwich, CT might be a good option if you can get him there. They specialize in language based LD so could probably meet his needs. Might be worth a phone call at least see the if they can make any recommendations in pNYC.
I have had less than good experiences with schools that market themselves as being designed for 2e students. I have also found hidden gems that serve this population - have gifted and sped programs that can overlap - but don't shout 2e so not necessarily easy to find.
We are now in our second year of non-typical programming. Not really homeschooling but instead a combination of school and nonschool programs that meet her 2e needs. One day a week at a nature program that uses experiential learning to explore science, agriculture and environment. 2 days a week getting 1-1 special ed services (paid for by our school district) in areas like speech, assistive technology and math. And 2 days a week taking regular academic classes at Fusion Academy - a chain of schools that offer 1-1 classes in a sensory friendly environment. (There are branches in NYC but don't start until 6th grade.) This gets the onus off of me trying to figure out how to go about homeschooling but still giving her all the flexibility that it would bring. Spaghetti and aeh do a great job above spelling out the benefits.
So I guess my advice is keep turning over rocks and you may find the school you are looking for. Contact those that seem close but might not work due to locatation or grade level to see if they can recommend an option that could work. Stay flexible. It might be that homeschooling could work but you don't want to forget about remediating the disabilities. You might be able to bring in a program from Lindamood Bell to help with that but it might be hard to do yourself without training.
I hope some of this helps. Good luck! I know there are some powerhouse folks here from NYC so hopefully they will chime in or pm you with specific recommendations.
