The spirit of the law as passed was to stop cable companies capturing and rebroadcasting over-the-air TV signals without compensation. So it shouldn't be too surprising that the Supreme Court made the judgement they did.
It's quite simple: when this law was created, was Aereo's use case considered? If the answer is 'no', then you can expect the law to be refined to cover it at a later date. Odd legal loopholes don't get to stand just because they've been around for a while.
Also, how about applying the spirit of the Statute of Queen Anne. Wouldn't like that one, I'm thinking.
"Spirit of the Law" - humbug.
We don't, no. The Supreme Court does, however.
On if such a performance was private or public: the decision said that it was a moot point as the "commercial objective" of Aereo was the same as if it were public. The law as it stands did not intend to separate private and public performances.