The two examples you have listed use 'while' to show a paradoxical condition: not wanting to admit fault but knowing you must be at fault sometimes, and not wanting big government but wanting the programs that big government brings. In both these examples, the modifiers are adverbial (not noun) and refer back to the preceding clauses.
To answer your doubts:
1-The structure (from a meaning standpoint) requires the 'while' modifier: They denounce big government while supporting specific government
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To answer your doubts:
1-The structure (from a meaning standpoint) requires the 'while' modifier: They denounce big government while supporting specific government
...