This is spot on, but this is still a PITA for everyone involved. There should still be a better way.
Yes, I fully expected this to be an article about how to "fix" this problem situation. I have to say that I have been in a few of the positions in this sort of chain, and now if I'm the inviter, I always just CC both. Not forwarding on the entire chain feels dishonest. (Obviously the invitee doesn't need to read the whole thread, but it's there for posterity, and to give context.) If the "busy person" is so rude that…
Consider the case of a friend looking for a job. If they come to me asking for an introduction, they may be open with me about things they wouldn't tell a stranger in the first minute of contact. E.g., concerns about the target company, fears they might not measure up, things going on in their life. It's hard for me to know what they don't want shared, I want to minimize the risk a violation of trust, and I don't want to guess at what the recipient might have an issue with. So I'll always write my own intro or ask for something I can forward.
Also, if someone dumps some unwelcome request from a pal in my lap, I'd be irritated that I had to do the work of bowing out. It's their friend, not mine, so it shouldn't be my problem. For that reason, I always ask permission to do an intro unless I have strong reason to believe the intro would be welcome. (E.g., if somebody told me they're looking for developers, I'll do intros without confirming an interest in each specific person.)