When Katie Bell was cursed we see this scene take place.
Professor McGonagall, looking alarmed as she
took the necklace from Harry. “No, no, Filch, they’re with me!” she added hastily, as Filch came shuffling eagerly across the entrance
hall holding his Secrecy Sensor aloft. “Take this necklace to Professor Snape at once, but be sure not to touch it, keep it wrapped in the scarf!”
Right before this Hermione and Harry state they need to bring the necklace to madam Pomfry, but McGonagall sends it to Snape.
“We need to get you up to the school, sir. . . . Madam Pom-
frey . . .”
“No,” said Dumbledore. “It is . . . Professor Snape whom I
need. . . . But I do not think . . . I can walk very far just yet. . .
“Right — sir, listen — I’m going to knock on a door, find a place you can stay — then I can run and get Madam —”
“Severus,” said Dumbledore clearly. “I need Severus. . . .”
“All right then, Snape — but I’m going to have to leave you for a moment so I can —”
Here we see Snape is who Dumbledore requires when he is under the effects of Dark Magic.
Snape hesitated, and then said, “I cannot tell. Maybe a year. There is no halting such a spell forever. It will spread eventually, it is the sort of curse that strengthens over time.”
Dumbledore smiled. The news that he had less than a year to live seemed a matter of little or no concern to him.
“I am fortunate, extremely fortunate, that I have you, Severus.”
“If you had only summoned me a little earlier, I might have been able to do more, buy you more time!” said Snape furiously.
Through out the whole series we are shown that Snape is exceptional at handling dark magic, knowing counter curses for dark magic ect. So its unsurprising that during the quiditch match over all of the teachers present he would be able to swiftly recognize what was happening to Harrys broom, as well as know what counter curse was needed.
Mcgonnagal, or Flitwick could have figured it out as well, but not near as fast as Snape would. And we know Dumbledore was not at this match, as it was unusual he was in attendance in the next match.