Sacrificial magic does have an immediate protective effect.

Lily sacrificing herself to save Harry did have an immediate effect without anyone having to do additional magic - it shielded Harry from the Dark Lord’s Killing Curse.

“Voldemort laughed softly in his ear, then took the finger away, and continued addressing the Death Eaters. ‘I miscalculated, my friends, I admit it. My curse was deflected by the woman’s foolish sacrifice, and it rebounded upon me.”
_{^{- Harry Potter and the Goblet of Fire, Chapter 33 (The Death Eaters)}}

Harry was similarly able to ‘shield’ everyone by willingly dying for them. Her sacrifice also made it so the Dark Lord couldn’t touch Harry, which also happened ‘automatically’ without further magic being needed to invoke it.

“You all know that on the night I lost my powers and my body, I tried to kill him. His mother died in the attempt to save him – and unwittingly provided him with a protection I admit I had not foreseen … I could not touch the boy.”
_{^{- Harry Potter and the Goblet of Fire, Chapter 33 (The Death Eaters)}}

It’s unclear if Harry’s sacrifice also made it so that the Dark Lord couldn’t touch everyone else, but it did shield them all from the Dark Lord’s attacks. It’s this immediate effect that Harry refers to.

“You won’t be killing anyone else tonight,’ said Harry as they circled, and stared into each other’s eyes, green into red. ‘You won’t be able to kill any of them, ever again. Don’t you get it? I was ready to die to stop you hurting these people –’

‘But you did not!’

‘– I meant to, and that’s what did it. I’ve done what my mother did. They’re protected from you. Haven’t you noticed how none of the spells you put on them are binding? You can’t torture them. You can’t touch them.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 36 (The Flaw in the Plan)}}

No additional magic was needed to allow an intentional sacrifice to shield the person it’s done for.

Dumbledore’s charm was just to protect Harry at the Dursleys.

What Dumbledore did was create a charm that ensures that Harry was protected while at the Dursleys’. Lily’s sacrifice took effect before that, and Dumbledore didn’t have to ‘activate’ the entire sacrificial protection - he just did the part that protected Harry at the Dursleys’.

“For he has been better protected than I think even he knows, protected in ways devised by Dumbledore long ago, when it fell to him to arrange the boy’s future. Dumbledore invoked an ancient magic, to ensure the boy’s protection as long as he is in his relations’ care.”
_{^{- Harry Potter and the Goblet of Fire, Chapter 33 (The Death Eaters)}}

Dumbledore’s charm simply used the power of Lily’s sacrifice, which had already happened and left its effect on Harry, allowing Harry to be protected while at the Dursleys’ house. Dumbledore explains this to Harry - Lily gave Harry a lingering protection by dying to save him, and this is why Dumbledore chose to use the charm that protected him while at the Dursleys’ house.

“You would be protected by an ancient magic of which he knows, which he despises, and which he has always, therefore, underestimated – to his cost. I am speaking, of course, of the fact that your mother died to save you. She gave you a lingering protection he never expected, a protection that flows in your veins to this day. I put my trust, therefore, in your mother’s blood. I delivered you to her sister, her only remaining relative.’

‘She doesn’t love me,’ said Harry at once. ‘She doesn’t give a damn –’

‘But she took you,’ Dumbledore cut across him. ‘She may have taken you grudgingly, furiously, unwillingly, bitterly, yet still she took you, and in doing so, she sealed the charm I placed upon you. Your mother’s sacrifice made the bond of blood the strongest shield I could give you.”
_{^{- Harry Potter and the Order of the Phoenix, Chapter 37 (The Lost Prophecy)}}

The effect of Lily’s sacrifice existed before Dumbledore used it to create his charm - its effect is why he chose to use that specific charm to protect Harry.

Harry survived as a baby immediately after his mother was killed, so it can't be because of something that Dumbledore did later. It's supposed to be an automatic consequence of the circumstances his mother's death.

In an interview Rowling explains the requirements:

JKR: Don't you want to ask me why James's death didn't protect Lily and Harry? There's your answer - you've just answered your own question - because she could have lived - and chose to die. James was going to be killed anyway. Do you see what I mean? I'm not saying James wasn't ready to; he died trying to protect his family, but he was going to be murdered anyway. He had no - he wasn't given a choice, so he rushed into it in a kind of animal way. I think there are distinctions in courage. James was immensely brave. But the caliber of Lily's bravery was, I think in this instance, higher because she could have saved herself. Now any mother, any normal mother would have done what Lily did. So in that sense, her courage too was of an animal quality but she was given time to choose. James wasn't. It's like an intruder entering your house, isn't it? You would instinctively rush them. But if in cold blood you were told, "Get out of the way," you know, what would you do? I mean, I don't think any mother would stand aside from their child. But does that answer it? She did very consciously lay down her life. She had a clear choice.

ES: And James didn't.

JKR: Did he clearly die to try and protect Harry specifically given a clear choice? No. It's a subtle distinction and there's slightly more to it than that but that's most of the answer.

MA: Did she know anything about the possible effect of standing in front of Harry?

JKR: No - because as I've tried to make clear in the series, it never happened before. No one ever survived before. And no one, therefore, knew that could happen.

MA: So no one - Voldemort or anyone using Avada Kedavra - ever gave someone a choice and then they took that option [to die] -

JKR: They may have been given a choice, but not in that particular way.

_{The Leaky Cauldron, 16 July 2005}

The book was released on 21 July 2007, so the interview predates it.

According to this statement, it is necessary that the victim could have lived and chose to die. Now Voldemort would never have given Harry the choice to live, so clearly the requirements for a protection are not met here.

Actually, there is no evidence in the book that anybody would be protected by Harry's death, except that Harry says so. The sequence is this:

Voldemort was in the centre of the battle, and he was striking and smiting all within reach. Harry could not get a clear shot, but fought his way nearer, still invisible, and the Great Hall became more and more crowded, as everyone who could walk forced their way inside.

...

Voldemort was now duelling McGonagall, Slughorn and Kingsley all at once, and there was cold hatred in his face as they wove and ducked around him, unable to finish him –

(Molly kills Bellatrix)

Harry felt as though he turned in slow motion; he saw McGonagall, Kingsley and Slughorn blasted backwards, flailing and writhing through the air, as Voldemort’s fury at the fall of his last, best lieutenant exploded with the force of a bomb. Voldemort raised his wand and directed it at Molly Weasley.

‘Protego!' roared Harry, and the Shield Charm expanded in the middle of the hall, and Voldemort stared around for the source as Harry pulled off the Invisibility Cloak at last.

The yell of shock, the cheers, the screams on every side of ‘Harry!' ‘HE’S ALIVE!' were stifled at once. The crowd was afraid, and silence fell abruptly and completely as Voldemort and Harry looked at each other, and began, at the same moment, to circle each other.

(Harry and Voldemort talk)

‘Potter doesn’t mean that,' he said, his red eyes wide. ‘That isn’t how he works, is it? Who are you going to use as a shield today, Potter?'

(more talk)

‘You won’t be killing anyone else tonight,' said Harry as they circled, and stared into each other’s eyes, green into red. ‘You won’t be able to kill any of them, ever again. Don’t you get it? I was ready to die to stop you hurting these people –'

‘But you did not!'

‘– I meant to, and that’s what did it. I’ve done what my mother did. They’re protected from you. Haven’t you noticed how none of the spells you put on them are binding? You can’t torture them. You can’t touch them.‘

(DH)

So after Voldemort tried to kill Harry, he was still "striking and smiting". The word "smiting" means, among other things, "attack with deadly or disastrous effect". McGonagall, Kingsley and Slughorn are flailing and writhing through the air, their landing would at least leave them disoriented and easy to kill for the Death Eaters. Harry tries to protect Molly, but the Protego can't stop the killing curse Voldemort would most likely use. Instead the fact that Harry is alive distracts Voldemort, he doesn't cast against Molly. Then Harry and Voldemort talk, and Harry claims that Voldemort can’t touch them, although his attack on McGonagall, Kingsley and Slughorn was effective.

Just as Voldemort tries to make Harry doubt himself when he says "Who are you going to use as a shield today", Harry tries to make Voldemort doubt himself when he says "You can’t touch them."

If it just needs anybody to be willing to die to protect everybody else, surely there would be someone willing, and Dark Lords would have a very hard time.

So there was no protection, it was just Harry's bluff.

Edit

Just for added context, when Voldemort said "Who are you going to use as a shield today, Potter?" he didn't really want someone else to be a shield for Harry, because he remembers how that ended the last time:

"I miscalculated, my friends, I admit it. My curse was deflected by the woman’s foolish sacrifice, and it rebounded upon me." (GoF)