Alternator part of 29 point inspection?

#1
So just trying to find out if the Alternator is supposed to be checked in the 29 point inspection.
Took my Corolla to Toyota of Manhattan as I had to replace water pump, blower motor and drive belt.
I asked the Service Tech to do a Diagnosis as the one I gave him was a year old and I wanted to make sure nothing else needed to be fixed. I don't believe they even did the diagnostic.
I picked up my car and it was making the same sound as well as another sound when picked it up. I brought it back the next week as they had to replace another part they didn't have in stock. Later that day (the 2nd time I dropped of car) the tech calls to tell me that "it wasn't the part we thought it was...."it's the alternator".

So my question is why wasn't it diagnosed the first time I took the car in and do I have to pay for the repairs of the alternator.?
 
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