There is still a missing element here in the concept, the why. Why an impossible birth
Because that was what it took to get Jasmine's essence to earth. It's the same 'why' as why it was Dawn's blood that opened/closed the portal or why Buffy had to be the one to free the Master or why Angelus was the only one who could awaken Acathla.
The conversation with Skip explains all of that:
SKIP
You really think it stops with her, amigo? You have any concept of how many lines have to intersect in order for a thing like this to play out? How many events have to be nudged in just the right direction? (looks at Lorne) Leaving Pylea. (looks at Gunn) Your sister. (looks at Fred) Opening the wrong book. (looks at Wesley) Sleeping with the enemy. (beat) Gosh, (chuckles) I love a story with scope.
GUNN
No way. We make our own choices.
SKIP
Yeah, sure. Cheese sandwich here, uh, when to floss. But the big stuff, like two vampires squeezing out a kid?
ANGEL
Connor.
WESLEY
An impossible birth to make one possible.
SKIP
That's what the kid was designed for.
LORNE
(chuckles) To sleep with mother love?
ANGEL
To create a vessel.
SKIP
Look out. The monkey's thinking again.
ANGEL
Being inside a human makes it vulnerable, doesn't it? That's why it had to stay hidden. Why it needed to create something stronger to pour itself into.
and why for that matter Cordelia? Why not Fred? Why not Lilah? She could have turned either of them half demon.
It's not just S3 and 4 that you have to look at for the reasons why Cordelia was the one to bring forth Jasmine - for that answer, you how to go back to S1, when Doyle's visions were transferred to Cordelia.
Again, Skip pointed that out:
SKIP
You really think it stops with her, amigo? You have any concept of how many lines have to intersect in order for a thing like this to play out? How many events have to be nudged in just the right direction? (looks at Lorne) Leaving Pylea. (looks at Gunn) Your sister. (looks at Fred) Opening the wrong book. (looks at Wesley) Sleeping with the enemy. (beat) Gosh, (chuckles) I love a story with scope.
Now, whether that was an accident that Jasmine capitalized on or whether she saw an opportunity and took it, we know from Skip that Jasmine chose Cordelia and put everything else into place - sending Cordelia to Pylea proved to her how much of an asset her visions are and made her decide to keep them, even though they were killing her, which in turn made her agree to become part-demon so that she could still have the visions, which made her body strong enough to survive the ascension and to house and protect Jasmine's essence until it could give birth to itself.
An impossible birth (Connor) to create another impossible birth (Jasmine) which the product of the first impossible birth and a human-demon hybrid.