It is funny that they use singular "la ceinture" to the belts of a group
I have noticed this sort of expression in the French language. Another example,
"Nous mangeons une crêpe."
It's possible but unlikely in my humble opinion.
Does the politician saying that include him/herself as well?
Is this a common idiom?
Yes, it is a common expression.
The answer is "we all." Wouldn't all of us be equivalent?
I listened several times and it really sounds like 'les ceintures'. Did anybody else get that?