Possible plothole

Maybe they assumed. But it being a snake doesn't automatically mean it's a basilisk, maybe saaay three different snakes cause petrification as a default, and the treatment for their poisons is fatal if wrongly used, so treated the accidental petrifications would kill them.

So maybe Dumbledore and others did make the connection, but never fully.

_
How do you know if you're happy or sad without a mask, or angry...or ready for dessert

Here's the thing. The teachers could've worked out it was a Basilisk, but...


They had no idea on how to prove it since the monster was never seen in its attacks, and had no idea on where the Chamber was hidden.

Putting it together and proving it...as well as exposing the monster are two different animals. Obviously Tom Riddle, never brought the basilisk back after hagred was blamed and expelled thus allowing him to continue his wizarding education to help him attain dark Lord status...... something that would have been curtailed had he allowed the basilisk to continue its reign of terror in his (voldemorts) youth!
IMHO

Even though Slytherin was a parselmouth doesn't mean that the monster has to be a snake. There are a lot of evil monsters in the Wizarding World, and maybe more than the Basilisk that can petrify people. And since there are so few parselmouth in the Wizarding World, nobody heard it roam the castle, until Harry came to school. Hermione was clever enough to work out that the monster had to be some sort of a snake since Harry was the only one that could hear it.

Interesting that you mentioned Hermione working it out but not the staff. Hermione had one big advantage: Unlike the staff, she knew about the voice Harry was hearing. While knowing Salazar Slytherin was a parselmouth wasn't enough for them to figure out the monster was a basilisk, that knowledge added on to the knowledge of Harry being a parselmouth and hearing a voice that only he could hear (also the petrified victims) was enough for her to figure out the monster was a basilisk.