Well, Anonymous, I was oversimplifying it for the purposes of understanding the concept a bit, but your position is flat-out wrong.
Go read 17 U.S.C. 106. (http://www.copyright.gov/title17/92chap1.html#106) There you'll find that the author of "literary, musical, dramatic ... or other audiovisual works" has an exclusive right to "reproduce the copyrighted work" (106(1)) and to "perform the copyrighted work publicly" (106(4)). Fair use does apply to certain "fair" uses such as for a non-commercial scholarly purpose, but in determining whether a use is fair the factor of "the effect of the use upon the potential market for or value of the copyrighted work" must be considered. If I wrote my play and I set the script out for you to view, you could take pictures for an investigatory purpose (my ex wants to claim it as part of her part of her property in divorce) or copy the pages for a critical purpose (a literary critic wants to release an article prior to the performance of the play). As it would directly impinge upon the potential market, any unauthorized production of the play would negatively affect my potential market, and would not be a fair use. (See 17 U.S.C. 107(4).)
Your arguments regarding the 9th amendment are difficult to understand. Are you suggesting that because the right to copy is not enumerated in the constitution, that we all have the right to copy whatever the hell we want? Congress and the courts seem to disagree with your view. Specifically for the Aereo case, I think you'll have a hard time arguing that the retransmission of video is an unenumerated right protected by the 9th Amendment. The people in the days of George Washington et. al. didn't have their eyes fixed upon their T.V.'s on the weekend, you know...
I'm not offering legal advice, but from a quick look it appears that Aereo's activities might be protected under 17 U.S.C. 111(a)(3) or another part of that section. That strengthens my view that Aereo will win.