I'm confused at what you are trying to say. As far as I can tell, it appears you are saying something like "don't quote religious texts in a way that makes them look like they have any claim on moral authority". And it appears to me that you are saying "moral authority", in this case, is a-religious and should not be related to any particular religion.
Well, fundamentally moral authority is a misguided idea, morals don't flow from an authority, morals are constructed and transformed over time by society. Taking your morals from an authority directly is itself immoral, as they don't have a foundation of human rights or empathy but rather are based on arbitrary ancient writing of arbitrary ancient writers.
I'm saying quoting religious texts without religious context (like you say, morals are not inherently religious) is not a secular act and can not be separated from the religion because of the history around quoting religious texts. Quoting religion in a moral discussion introduces religion into a discussion where religion is not necessary, and could even be harmful.
That's not to say you should never do it, but doing it is not the same as quoting any other text or other author. Again I'm not sure why this is controversial, regardless of the other comments quoting Hitler or Mao's red book in a moral discussion would very much be very weird if there is no reason to introduce any of his writing.
> Taking your morals from an authority directly is itself immoral, as they don't have a foundation of human rights or empathy but rather are based on arbitrary ancient writing of arbitrary ancient writers.
Where do you think should morals come from? A more precise question would be "where does human morality arise from?" Or even more fundamental, "why is it wrong to do one thing and not another?".
Given what you've said about religion and moral authority, I am genuinely curious to know your answers to these questions.
However, its quite simple to resolve. The gist of it is: if you need a deity holding judgement over you to distinguish right from wrong, you are probably a psychopath.
Am I getting this right?