I can easily see it resulting in the complete dissolution of copyrightable open source, simply by labeling each individual transmission of the work a so-called "private production" or whatever is necessary to somehow make it inapplicable to being an infringement.
Performing a work privately isn't infringing, because the exclusive right of performance under section 106 only applies to public performances. However, unlike the GPL, this does not 'infect' the work or any instance of the work. Copyright applies to the work as much as it ever did. Therefore, if you privately perform a work, by, for example, watching it on TV, you cannot make a copy of it, distribute copies of it, or make derivative works based upon it. There might be a fair use exception to certain acts depending on the circumstances (e.g. copying a work by recording it on a DVR, whether this is done at home or remotely over a network), but fair use can apply to any kind of use to any kind of work, so long as the use is fair. There's nothing special that a holding against Aereo would do vis a vis fair use and its applicability to open source software.
You're worried about things that have no chance of happening, possibly due to ignorance of US copyright law. It's not just making a mountain out of a molehill. You're making a mountain out of sheer imagination; there's no molehill or any other thing to enlarge in the first place.