So why would he do everything he could to raise his son to want to grow up to be a Death Eater?

He wants him to grow up to be the leader of the Purebloods, which exactly the same thing as leader of the Death Eaters.

"Ignore him," Madam Longbottom said, the instruction addressed to Harry as well as Neville. "He must spend the rest of his life pretending, for fear of your testimony under Veritaseum."

And how would that constitute evidence that he was not a willing Death Eater if everyone believes he is faking? Certainly he should never affirmatively say he was a willing Death Eater, but this little speech that no one believes amounts to evidence that he wasn't a Death Eater to who? The speech does him absolutely no good.

In fact, I'd say the speech only harms him. The Longbottoms of the world won't believe him, but those who sympathized with the Death Eaters, much of his natural power base, would likely be annoyed at the disavowal, even if they didn't believe it either.

When a schemer says something that does him absolutely no good, a reasonable interpretation is that it is true.

And I agree that it's incongruous for Lucius to speak to Voldemort "in the spirit of friendship", I just think it's more incongruous with the Imperius narrative.

As pointed out, he needed Lucius to draw Purebloods. First he asked. When Lucius declined, he spelled him. He forced compliance. Not so friendly, but the least abusive path that achieved his goal. That might count as friendly with a Flawless Instrument of Death.

I expect we get the backstory on Lucius filled in some day. Although I'm wondering if EY has made a major change in the plot. Seems like a lot of work went into the Draco character, but he seems like a soap opera character who hit it big in the movies, and was quickly written off the show.