@MontereyJack,
Using "regard" is
not incorrect in the sentence, and using "regarding" would be incorrect in the sentence unless you change the meaning of the sentence, but as it stands the grammar is correct. If you simplify the sentence it becomes clearer.
Correct:
That politicians wield power but
regard politics as the decisive influences in society betrays a misunderstanding of what makes human beings tick.
Incorrect:
That politicians wield power but
regarding politics as the decisive influences in society betrays a misunderstanding of what makes human beings tick.
This is because you don't say "The fact that politicians regarding..." and that is what the initial "That" does to the sentence. Drop that word and yes, you need the gerund. Examples:
Correct:
Politicians wield power but
regarding politics as the decisive influences in society betrays a misunderstanding of what makes human beings tick.
Incorrect:
Politicians wield power but
regard politics as the decisive influences in society betrays a misunderstanding of what makes human beings tick.
Simplified even further might make it even clearer.
Correct:
Politicians
regarding it as decisive betrays a misunderstanding of humans.
That politicians
regard it as decisive betrays a misunderstanding of humans.
Incorrect:
Politicians
regard it as decisive betrays a misunderstanding of humans.
That politicians
regarding it as decisive betrays a misunderstanding of humans.
Quote:It's not what politicians regard, but what the the regard of everybody is or should be.
If that meaning is desired you need to change the sentence, yes, but I'm not commenting on the ideal meaning (and on a side note I'm not sure how one meaning is "right" and the other "wrong" from an English point of view, both are workable sentences), but rather the grammar. The sentence is grammatically correct as is with "regard" attached to politicians.