Because they had staked their reputations on a public statement that got picked up by the media across Europe and North America. If they had to admit they were completely wrong, that Amanda, Raffaele and Patrick had nothing to do with it and it was all this other guy, Rudy Guede, they'd look like fools. Reporters would ask, "why did you make this dramatic announcement before you even looked at the fingerprint and DNA evidence?"
Once they realized they had screwed up, they went into damage control mode. They had to release Patrick, because he had a multiple witness alibi. So they substituted Rudy for Patrick in the "sex game," and everything else stayed the same. That way they could say they were right all along, and blame Amanda for tricking them into making a false arrest.
Next, they got busy with evidence. They couldn't find any. So they made up stories... "We found bleach receipts!" "We found the book Amanda said she was reading at Raffaele's place, but it was at the cottage!"
Things looked bad when Raffaele's family did the spadework to prove the bloody shoe prints didn't match his sneakers, and announced it on TV. But, lo and behold, the very next day... they found the bra fastener. How about that? Just when it looked like they weren't going to have any proof of Raffaele's involvement, they found exactly what they needed, in a pile of junk that had gotten swept into a corner.
I followed this case for quite awhile before I formed a strong opinion. But at a certain point, it became very, very obvious, to me at least. That is because I have read about many other cases like this. It's a particular interest of mine.
See above.