He did try to carry out the plan all by himself.
Remember all those attempts that failed? The necklace that almost killed Katie Bell, the poison that almost killed Ron, ... That was Draco trying - perhaps rather half-heartedly, as Dumbledore himself surmised - to carry out the assassination. No Death Eaters would have been required if either the necklace or the poison had successfully reached Dumbledore.
"Oh yes, I do," said Dumbledore mildly. "You almost killed Katie Bell and Ronald Weasley. You have been trying, with increasing desperation, to kill me all year. Forgive me, Draco, but they have been feeble attempts ... So feeble, to be honest, that I wonder whether your heart has been really in it."
He was scared and wanted backup.
Assassinating Dumbledore is a big ask, regardless of the fact that in theory he could do the deed quietly in a corner. Dark Mark or no Dark Mark, Draco is just a teenager, and Dumbledore is one of the greatest wizards of the age. If it came down to a straight fight between them, Draco would have no chance. He needs to either catch Dumbledore by surprise or be one of a larger group all attacking at once. (In the end, he was lucky beyond his wildest dreams to find Dumbledore greatly weakened and apparently alone.) You make it sound as though "corner Dumbledore and kill him" is simply one of an A-B-C sequence of things to tick off a to-do list, but in reality that's one of the toughest tasks any Death Eater has been given.
"Draco, Draco, you are not a killer."
"How do you know?" said Malfoy at once.
He seemed to realise how childish the words had sounded; Harry saw him flush in the Mark’s greenish light.
"I see," said Dumbledore kindly, when Malfoy neither moved nor spoke. "You are afraid to act until they join you."
In addition to the difficulty of the task, there's Draco's own nerve to be taken into account. Does he really have it in him to murder his headmaster in cold blood? Death Eater backup would help to "screw his courage to the sticking point" and ensure that he actually carries out the task. We see how his nerve fails when he's talking to Dumbledore alone but he seems closer to actually doing it when he has the others breathing down his neck. Maybe he didn't fully trust himself not to turn tail and flee without companions who were more ruthless and vicious than he.
"My options!" said Malfoy loudly. "I’m standing here with a wand – I’m about to kill you –"
"My dear boy, let us have no more pretence about that. If you were going to kill me, you would have done it when you first Disarmed me, you would not have stopped for this pleasant chat about ways and means."
As it turns out, even when he does have the other Death Eaters egging him on, he's hardly able to perform the deed even before Snape shows up to do it for him:
Now, Draco, quickly!' said the brutal-faced man angrily.
But Malfoy’s hand was shaking so badly that he could barely aim.
"We’ve got a problem, Snape," said the lumpy Amycus, whose eyes and wand were fixed alike upon Dumbledore, "the boy doesn’t seem able –"
All quotes from HP and the Half-Blood Prince, Chapter 28.