As I already stated, the biggest advantage is "access." Having a big name politician on your board means you can have them chat with their friends in congress about why they should vote yes/no on a certain bill. It helps the company convey their opinions to the people who matter. There are other advantages, like increasing the organization's prestige, and having access to someone with a great deal of knowledge about the political process (who might be persuaded, who is a lost cause, etc.)
It seems that politicians make a lot of decisions that aren't socially optimal because they want money from lobbyists and other campaign contributors. Presumably, the purpose this money serves is to keep them in office by allowing them to advertise a lot the next time they're up for reelection.
So the question then becomes, "why do they want to remain in office?". I could think of two reasons: money and power. From what I know, politicians have a pretty high salary (congressmen make ~$175k), so that's an understandable motivator. But power is the one I don't understand.
Supposedly they want to remain in office so they could use their power to have an influence. I don't know too much about politics, but it seems that politicians spend most of their time catering to lobbyists and voters rather than pushing the things they actually believe in. So much so that they aren't actually exerting that much power. And it seems that most of this catering is to special interests and is socially suboptimal. (I may very well be wrong on these points. I really don't know but it's the impression I get.)
Why are congressmen so motivated to stay in office, make $175k a year, exert a minimal amount of real power, and spend their time catering to lobbyists and making socially suboptimal decisions? I'm sure they could make twice as much in the private sector. I feel like there's something obvious that I'm missing here, but I'm genuinely confused.