You are overlooking the word “improperly”.
The section deals, first of all, with using influence to further somebody's private interests or the private interests of their friends, etc., and then, when you're talking about unrelated third parties, it is “improperly” furthering them.
Obviously, part of your job is to further the interests of your constituents, etc., but to do it properly and not improperly. That's the distinction in the act.