Draco DID want to be a Death Eater (and here’s why)
If I had a Sickle for every time I’ve read that Draco became a Death Eater against his will, forced by either his father or Voldemort, I’d be as rich as the Malfoys. However, that is not true in canon, and Draco is much more compelling and tragic for it.
It’s explicitly shown in HBP that Draco was enthusiastic about serving Voldemort in the beginning. Bellatrix, who’s ever eager to call out any unfaithfulness to Voldemort, defends Draco:
“And I will say this for Draco: he isn’t shrinking away from his duty, he seems glad of a chance to prove himself, excited at the prospect —” (HBP2)
Draco himself gloats about it:
“Well, you never know,” said Malfoy with the ghost of a smirk. “I might have … er … moved on to bigger and better things.” [...] “When the Dark Lord takes over, is he going to care how many O.W.L.s or N.E.W.T.s anyone's got? Of course he isn't. It'll be all about the kind of service he received, the level of devotion he was shown.”
[...] Crabbe and Goyle were both sitting with their mouths open like gargoyles. Pansy was gazing down at Malfoy as though she had never seen anything so awe-inspiring.
“I can see Hogwarts,” said Malfoy, clearly relishing the effect he had created as he pointed out of the blackened window. (HBP7)
And he’s preoccupied with the “glory” he thinks he’ll get by completing his mission:
“I know what you're up to! You want to steal my glory!” (Draco to Snape, HBP15)
“[Snape]'s been offering me plenty of help — wanting all the glory for himself — wanting a bit of the action — [...] But I haven't told him what I've been doing in the Room of Requirement, he's going to wake up tomorrow and it'll all be over and he won't be the Dark Lord's favourite any more, he'll be nothing compared to me, nothing!” (HBP29)
Of course there are threats and fear involved, as well, since this is Voldemort we’re talking about, but it’s both the carrot and the stick. When Draco starts to think he might fail, he focusses on the threats, hence:
“No one can help me,” said Malfoy. His whole body was shaking. “I can't do it... I can't... It won't work… and unless I do it soon... he says he'll kill me…” (HBP24)
Nonetheless, as per the previous quotes, he oscillates between being terrified of failure and chasing the rewards of success up until the very end, in the Astronomy Tower.
As I've argued extensively in my Hand of Glory meta, I see Draco’s becoming a Death Eather as an attempt to both prove himself to his father and to prove himself better than his father.
It’s not that Draco has lost love or respect for Lucius, but he still wants to take the opportunity to make his father finally see his value by out-doing him. If Lucius’s DoM blunder triggered the Malfoys’ fall from grace, Draco’s success will earn them even more honour than they had before.
At the same time, though, Draco’s actions are not truly emancipatory because his father remains the point of reference that determines his worth.
Ultimately, Draco’s motive for taking the Mark is less about belief in the cause, and more about his daddy issues paradox: wanting to prove himself a grown up man, but doing so in a desperate, rash bid for the paternal validation he so sorely lacks.
Draco starts HBP insisting that he’s “...not a child, in case you haven't noticed, Mother”, and “perfectly capable of doing [his] shopping alone” (HBP6), but by the end of the book he’s feeling quite incapable of doing things alone, and still struggling with his need for approval from a father figure.
To me, this is much more interesting than simple external coercion. Draco’s own lack of independent self-worth is what leads him to destroy his life, and what renders him unable to be dissuaded from it. He dismisses anyone who tries to warn or help him, because he assumes they must share his own repressed lack of belief in himself, and marches solitary and obstinate towards his own ruin.
