This is a great point Semil.  I find it so much easier to help people who are very specific with their requests.   For instance, I'm happy to help recent grads find jobs, but so many of them answer the question of "What do you want to do?" with "Anything -  I just want to get into startups" and that is the worst answer.  If someone is specific, it is so much easier to refer them to someone in their chosen area that can connect them to others.  If I can potentially refer someone to everyone I know, I'll probably refer them to no one, but if I can think of 1-3 people who would be valuable, I'm happy to refer them.  People need to realize more opportunities come from traversing the relationships of others than simply being open to anything.  In your example, founders should find the 3 things they most value in an investor/potential board member and then ask for recommendations before asking for introductions.

If at all possible, the person asking for the introduction should explain why it would be interesting for the other person to talk with them, too.

The best cases are when the meeting benefits both people who are meeting.

Yes, but I'm being very specific about investments. A founder just wants to meet "Fund X Capital" because it's on his/her list, but can't name a partner there, let alone which one would be relevant and why. This all breaks down because investors often want to be seduced by a potential investment and founder, and this lack of targeting and precision kills the buzz -- and time.

Yes, when we finally hit a person that makes sense, I'll write: "Great, now send me an email I can forward to Person X."

Your point is a good one.

Which fund matters less than which partner.

The expertise and investment thesis of the partner better match the startup.

If it doesn't, the introduction makes no sense.

Some research up front will save everyone a lot of time.