If I were really to invite 5 of my friends, their most likely response is:
a) not to join at all
b) not to share (that's extra work)
c) or being generous: invite one or two people
But let's roll with the post's bizarro assumptions that each accepted invite results in 2 more and see what happens if there are just 6 other entrants initially (this problem favors your argument with less entrants, not the other way around).
Your chances without sharing (remember, those other entrants are going to be able to bring in a billion persons each):
1/(1 + 2 * 6 * 2^30)
Your chances with sharing (you get two extra entries):
(1 + 2)/(2 * 7 * 2^30)
That's right. Every human being alive has joined. Only 6 other people started in the contest and you still improve your chances 157% by sharing.
The absolute only case in which it makes sense not to share is when you're the only person in the contest. And I would hope most people know not to try to improve on 100% odds.