Depends on the morality.
"We acknowledge that morality is beneficial to society, but are there advantages to a lack of morality?"- Critical-Tim Depends what one considers an advantage.
"How does The Greater Good align with morality?"- Critical-Tim Depends what one considers the Greater Good, or morality.
"Why is The Greater Good not always considered moral if it is the greater good? (This would imply what we consider moral can be the lesser good)"- Critical-Tim Because there are different moralities.
"Why are some things considered immoral, but as a whole more beneficial to the world?"- Critical-Tim Because there are different moralities.
You said not to mention a religious perspective,
But I will anyway, as it need not be.
My religious family members define immorality as hurting people,
Morality as doing what is right, helping people, I suppose.
I'd suppose for many people the purpose of morality is to 'be moral,
Circular I suppose,
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Though I don't define morality as doing right by other people,
I often use that definition in conversation.
Common definition.
. .
'Reason for that definition,
You can break a single arrow, easier than a bundle.
A house divided, cannot stand.
. .
Reason it even applies to outsiders, often,
Is that dehumanize others,
Can be easier to dehumanize parts of one's own society.
Though many societies exist 'based on 'only applying morality to one's own group.
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If I encourage other's to do right to others including me,
But secretly do wrong myself,
I can enrich myself financially in this manner,
. .
But possible I will suffer emotionally, mentally,
If I believe this is wrong,
Or because of my own actions, cannot trust others,
Cannot 'like others.
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Course, depends on definitions of suffer, or doing 'Right/'Wrong are.
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Greater Good,
'Can contrast with smaller goods,
If smaller goods are not viewed as 'less,
As not mere means to ends, of the Greater Good.