Your textbook is wrong in saying that  is wrong. Whenever
Q work in the opposite direction towards expectation of
P ではないが Q is okay. For example, it is fine in the following context:
'Is there any Asian?'
'I am a Japanese.'
C: 僕は日本人ではないが, シンガポール人です。
'I am not a Japanese, but am a Singaporean.'
In this context,
僕は日本人ではない closes up one possibility of C being an Asian, but then,
シンガポール人です opens up another possibility, working towards the expectation of, and in fact affirming, C being an Asian.
Usually, you read as it is written. If you want
じゃなく, you will usually write so. But it is not that wrong to read
じゃなく. It is wrong/correct to the extent that reading English
Actually, even when
Q do not work in the opposite direction, you can use
PがQ as long as
P is providing a new topic into the discourse.
'He passed the examination, and that is due to his consistent effort.'