I also live in New York. If NYC won’t allow you to remove the tree I guess that means the tree is on City property not yours? I know where I live a City owned tree has to be ¾ dead before the City will even consider taking it down.
I also have a big old oak tree on my property right next to the side of my house. It has been there for 100 years. It was half the size when we purchased our house over 40 years ago.
I have been told by actual tree people that oak trees roots grow straight down not like maples that branch out all over. We have never had any plumbing problems in all the years we have lived here. We have had this tree professional trimmed over the years at around $1000 a time. but as you know oak trees grow back from the trim really fast.
I did get some quotes and most prices are around $12,000 to cut down this big oak.
In your case if you don’t own the tree you might be able to go to the courts to petition for the tree to be trimmed by saying it is a hazard to your personal property.
If you actually own this tree you have to know what you are doing when it comes to trimming or cutting down a huge tree. use professionals! look in the yellow pages, pennysaver or even get recommendations from people who have trimmed or cut down a tree. Dont go it alone.