We don’t know that Snape wanted to eliminate bullies. Snape’s intervention in SPEW battles caused a serious escalation in the conflict, but it was Quirrel’s intervention in the final battle that continued the escalation to the point where something had to be done to stop it. We do not know what Snape’s intention was for that final battle.
He might not have had an intention for that battle in the first place:
And so it was with some puzzlement, a few days later, that Daphne looked at the parchment delivered to her at lunch, drawn in a hand so shaky it was almost unreadable, saying:
2 this afternoon at the top of the stairs going up from the library REALLY IMPORTANT everyone has to be there—Millicent
Daphne looked around, but she couldn’t see Millicent anywhere in the Great Hall.
“A message from your informant?” said Hermione, when Daphne told her. “That’s odd—I didn’t—”
“You didn’t what?” said Daphne, after the Ravenclaw girl had stopped in mid-sentence.
The note from Millicent was sent by the bullies; Snape didn’t send a note at all.
We don’t know that Snape wanted to eliminate bullies. Snape’s intervention in SPEW battles caused a serious escalation in the conflict, but it was Quirrel’s intervention in the final battle that continued the escalation to the point where something had to be done to stop it. We do not know what Snape’s intention was for that final battle.
He might not have had an intention for that battle in the first place:
The note from Millicent was sent by the bullies; Snape didn’t send a note at all.