I don't think it's even as strong as this. I doubt there was any communication whatsoever between the US DoS and the Italian judiciary, nor that there was any communication from US DoS to Italian Executive Govt to Italian judiciary. I do however think that the SC judicial panel would have been minded of the US's stance on extradition, and that this might possibly have prompted them to consider the logical ramifications: is the US DoS saying the US wouldn't extradite simply because they are blindly protecting a US citizen regardless of the evidence of guilt, or is this instead a useful indicator that convictions would be wrongful here?
I think we have to bear in mind that it's a cornerstone of justice systems in modern democracies that 1) everyone is equal under the law, 2) nobody can buy or otherwise influence the administration of justice, 3) the judiciary must be - and must be seen to be - wholly independent from every other arm of government.
And that's why I would argue that if this SC panel of five judges really was "nobbled" (and I think it's a preposterous and incorrect accusation, but for the sake of this argument...), then observers - especially Italian ones - should be focussing their attention pretty heavily on the shocking thought that the Italian judiciary, from the very highest level down, can be bought, influenced, corrupted etc.
I feel confident that if there was ever any credible thought in the UK that our Supreme Court or High Courts (or even Crown Courts of Magistrates' Courts) could be corrupted at will, or bought off, or politically influenced, there would be a stupendous outcry at a number of levels. In fact, it would trigger something of a constitutional crisis here, I'm sure.