it would make the most sense to optimize the homepage for 'NYC apartments'
I agree.
If I may ask, why do you desire to rank so well for "Platinum Properties"? If that is your company name, most people who search for you will find your site whether you are the first result or the tenth. I would recommend focusing on the other two pages you suggested.
Home page: NYC apartments and another variation of the phrase such as "apartments New York" or whichever key word phrase you feel would be best.
Page1: NYC Apartment Rentals and another variation of that phrase
Page2: NYC Apartment Sales and another variation of that phrase
For the title tag, any change can affect your current ranking. You can make no change and your ranking can move up or down based on a variety of factors ranging from Google changes to how your competitor's rankings change.
Have you considering moving your company name to the end of your title tag? You would want to improve the first half but a fast example would be
<title>NYC Apartments Rentals and Sales | Platinum Properties</title>