By arriving at the scene, Harry certainly made a very clear pathway for extracting the stone from the mirror. We can't know for sure what might have happened had Harry not been there. Perhaps the mirror would have prevented Voldemort from getting the stone altogether -- Dumbledore seemed to think it was effective protection -- or perhaps not. Voldemort can be very determined, after all.
To answer the title question, though: yes, Harry and co. did save the day: Voldemort did not get the stone, and in fact, was set back a few paces, directly as a result of the actions of Harry and friends. But also yes: they very nearly did lose the day. Things could have gone badly if Harry never had a chance to discover the effect his own skin could have on Quirrell.
As to the final question - "were the adults correct all along that 'our heros' were meddling outside their competence?" - it's worthwhile to point out that perhaps not all adults held this view. Dumbledore, in particular, might have specifically intended Harry to get to the final showdown. As we see Harry ruminating at the end of the book:
"D'you think he meant you to do it?" said Ron. "Sending you your father's Cloak and everything?"
"Well," Hermione exploded, if he did – I mean to say – that's terrible – you could have been killed."
"No, it isn't," said Harry thoughtfully. "He's a funny man, Dumbledore. I think he sort of wanted to give me a chance. I think he knows more or less everything that goes on here, you know. I reckon he had a pretty good idea we were going to try, and instead of stopping us, he just taught us enough to help. I don't think it was an accident he let me find out how the mirror worked. It's almost like he thought I had the right to face Voldemort if I could...."
This vaguely resembles Dumbledore's own words when he later on reveals information to Severus Snape in the Deathly Hallows:
"We have protected him because it has been essential to teach him, to raise him, to let him try his strength," said Dumbledore, his eyes still tight shut.