They want people to stop sharing RIAA-licensed music over the Internet, or in the alternative, they want those who do share RIAA-licensed music to pay them for the privilege. Like it or not, that part really isn't so unreasonable. Unfortunately, the other thing they want is a monopoly in music publishing and distribution, and they're happy to use over-reaching, heavy-handed tactics to get it. That part is completely unreasonable.
There's a real lack of understanding on the RIAA's part -- they stubbornly refuse to acknowledge that the world has changed, that production and distribution of music now is so cheap and easy that you no longer need to be annointed by them or represented by them to create, record, produce, and distribute music. They still see themselves as the gatekeepers of the music industry. And the fact that a huge volume of RIAA-licensed music is in fact traded illegally over the Internet feeds their belief that they deserve to recover licensing fees by any means necessary.
Also, their tactics have worked in the past to help them control other media. This covenant not to sue scheme seems an awful lot like a blank media surcharge to me. A blank media charge combined with RIAA-mandated DRM worked to effectively kill digital audio tape (DAT) for recording purposes, and I understand that they get some sort of blank media charge on at least some CD-R media, so they won in both cases.