The Staten Island dump was the initial moving place for all of the debris. And, it had been cleared of everything within a couple years.
When I went in 2009 back to New Jersey for a visit to family, my mother took me up to Mt. Mitchell (between Atlantic Highlands and Highlands), to show me the 9-11 Memorial that had been erected there with the names of people from the area who had perished that day. On each side of the monument there was a part of the beams from the Twin Towers on each side. It was very moving. All of this had been done since my leaving New Jersey in the summer of 2003.
There is a cross at Ground Zero that was erected from the beams of the WTC.
Other beams have been been sent across the country to various communities for memorials; firetrucks in different areas have been mounted with pieces of the beams. The Naval Ship bow - USS New York, was created by melted remains of some of the beams. It was commissioned for active duty in August of 2009.
Yes, I agree that the burial at sea of any and all soil and such that would hold the remains of those fallen that day is appropriate.