Public School or Continued Daycare

No, leave her to get all of the nurturing and attention at the day care setting. After she gets the strong foundation from the day care, then she'll graduate and get skipped when she goes to public school.

You may be better off leaving her in daycare for now.
The school may try to hold off on letting her begin kindergarten because of her age. The birthday cutoff for NYC is December 31st, meaning she must be 5 be 12/31 of the year she'll enroll.
An example of this arbitrary cutoff point is my daughter and niece, my daughter was born in October 1996 and my niece was born just 3 months later in January 1997. While my daughter will be graduating 5th grade this year my niece won't be starting 5th grade until the fall. NYC wouldn't let my SOL enroll my niece in kindergarten until she was almost 6 because she missed the birthday cut-off by about 2 weeks.
You may be able to get her in early if you put her into private school(if possible), I don't think they are as strict.

She's only 3! I would keep her in daycare for another year. Don't worry about the school wanting to skip her a grade -- they cannot do that without your consent. In such an early grade, I would advise against it, as kindergarten and first grade are as much about learning to get along with others as academics. Good social skills form a critical foundation for later learning.

well it depends on lots of factors...would it be finicially better to have her go to public school? Does the private daycare offer more hours? Will the pre-k be in the school she will attend for Kindergarten? It would give her a chance to make friends with her future calssmates...it would give you a chance to see the school she'll be at and get involved in the PTA/PTO/PA at the school. The will not skip her. It is all about the age in the NYC DOE. They are strict about that. I taught Pre-K in Brooklyn for 8 years and I think that it really helped the kids to transition to Kindergarten. I noticed later when I had Kindergarten that students that came from private centers at time had a hard time adjusting to the rigid format of pubic school. It really depends on what you are looking for. You can always request a visit to the Pre-K to see if you like it also. Keep us updated!

If you feel comfortable with the Daycare you should leave her there one more year. However, if she was accepted into a lottery type of public school you have to consider that she will not get another chance to get into the public school. You may have a hard time getting her into a good public school next year.