When Voldemort rescues Bellatrix at the Ministry, we could argue that he acts on instinct. He can’t let Bellatrix be captured by the Aurors and sent back to Azkaban. Not even for five minutes.
He does control the Dementors by then, but perhaps he was traumatised by losing his second-best Death Eater, who had been kissed by a Dementor only a year earlier. What if the Aurors ordered a Dementor to give Bellatrix the Kiss without a trial, just as they did to Barty?
A year later, he doesn’t send Bellatrix to infiltrate Hogwarts. Instead, he sends only lowly Death Eaters such as the Carrows, Rowle, Greyback, and Gibbon, who ends up being killed because of Rowle’s stupidity.
Why? Because Voldemort doesn’t think for a second that Draco Malfoy will actually manage to kill Dumbledore. He expects failure. He expects losses.
He sends his least valuable Death Eaters to Hogwarts. If they get arrested, he already knows he’ll free them a few weeks later along with the others captured at the Ministry. And if a Dementor kisses one of them in the meantime, he doesn’t care in the slightest. If one of them dies, like Gibbon, that’s no great loss either.
So, of course, he doesn’t send Bellatrix on such a risky and ultimately farcical mission.
Now, after coming face to face with Bellatrix’s mortality (he’s immortal, yes, but she isn’t), and knowing she’s his last and best lieutenant, it’s actually natural that he’d want a child with her to replace her (it’s a very utilitarian vision of Bellatrix, but this is how things make sense in his brain to keep control on his emotions).
A child he’d raise with Bellatrix to ensure her loyalty.
And why should the child be his rather than Rodolphus’s? Because there’s probably a great deal of delusion involved too. Delphini becomes a new obsession, a new project, one that allows him to avoid thinking about Bellatrix’s eventual death. Delphini is a new kind of Horcrux, now that he’s completed his oldest, most important task and ambition. He’s finally immortal… now what?
Delphini is his way of coping with his abandonment issues. He won’t lose Bellatrix through betrayal, but he will lose her to death. And as we know, losing her will unravel him to the point that his magic explodes with the force of a bomb. That uncontrolled surge of magic only shows that both his fear of her death and his attachment to her are completely repressed and largely unconscious.
His fear of losing Bellatrix, of being abandoned, is the reason Delphini’s birth makes sense.