Those are some good points you have, and The Met is excellent, and accessible to all. Many people make up the difference with The Met, so cost should never be a barrier to entry.
You can do all kings of fun things in Central Park, perhaps have a picnic lunch in Sheep Meadow, take a walk to search for the statues, go to Belvedere Castle, etc. Depending upon the season, there's the Victorian Amusements and/or ice skating and, of course, Central Park Zoo. The Zoo is $3 a head for 12 and under, $8 for adults. And, there are also free guided tours of Central Park. Perhaps they have special events for Scouts? I am not sure if they do or not, but it might be worth an inquiry, since I know that groups should call for a custom tour, and it's free. Here's a link to the Central Park Conservancy:
Central Park Conservancy: Walking Tours (http://www.centralparknyc.org/site/PageNavigator/programs_walkingtours - broken link)
There are many free walking tours that you can take around the city, sponsored by various business improvement districts.
Many of the museums, The Whitney, MoMA, and, Cooper-Hewitt, Folk Art, and The Guggenheim have free or pay-what-you-wish dontation programs on Friday evenings usually in the 4 or later to close time slots.
You could do something fun like get tickets to a taping of a New York based show as well. Here's a link for some information on that:
Discount Broadway Tickets and Free TV Show Tickets in New York City
You can visit the Stock Exchange, Rockefeller Center (not the NBC tour for free, I think it's $10-$12 a head, depending upon group size), the Cathedral of St. John the Divine, and St. Bartholomew's, and St. Patrick's Cathedral.
I don't know how into movies like "National Treasure" your Scouts are, but you could explore some of the sites in Lower Manhattan used in the movie, such as Trinity Church, and combine it with the Stock Exchange. Federal Hall National Memorial is also in the area and is a National Park. In Season, you can go to Governor's Island via free ferry as well.
In Upper Manhattan, General Grant Memorial (aka Grant's Tomb) is also a member of the NPS and has free tours and some great views over the Hudson as well.
You could visit the Sony Technology Wonder Lab on Madison Avenue as well. Here's a link:
Sony Wonder Technology Lab
Many NYC parks have some free attractions, including music and such in warmer weather. Here's a link to NYC parks for information, where you can search by free events as well:
New York City Department of Parks & Recreation
The New York City Public Library is free as well, which features some programs for various ages of children, and sometimes there are free events in Bryant Park too.
And, if I am not mistaken, I do believe that the Museum of Natural History also has a suggested admissions policy, for in-person, ticketing, not advance purchase. The fees for some of the shows, however, are fixed, but the admission to the museum is scalable, I believe.
American Museum of Natural History
The National Museum of the American Indian is part of the Smithsonian, and has a New York branch. All Smithsonian admissions are free, so it might be a good place to visit. Here's a link:
National Museum of the American Indian
Something that one of my excellent fellow New York board posters has suggested to people in the past is that in lieu of a double-decker bus tour that people take the city buses with their Metro Card on certain routes and take in the sights that way. You don't get the narrative, but you can see and experience the city that way as well.
There's a lot to do in NYC without breaking the bank, so it should be a fun trip for the Scouts.