My "can't miss" are: Times Square (but you'll see that with Beatlejuice). Metropolitan Museum of Art (it or the Louvre are the best art museums). Central Park. Top of one of the buildings (Empire State, etc). While I haven't done Statue of Liberty in decades, I'd put it on the list as well.

A final thought, but are you familiar with a "City Pass"? I've done variations of this in several cities including Chicago, Seattle, San Fran and Paris. Basically you pre-buy a pass that gets you discounts to major attractions, which is nice. But most important, IMO, you usually you get preferred access/get shorter lines.

A quick search online and New York seems to have the official "City Pass" and something called "The New York Pass." Both seem to have a number of the attractions on your list. Looking at these, the City pass let's you visit few attractions, but over a 9 day period. The NY Pass seems to work by day, but you can see more attractions (if you rush around???). Up to you, but the ones I've done had the City Pass format.