So, why didn't any of the Exeter's crew try beaming down to the planet after they were infected and people started dying?
The ship's medical officer had enough time to record a warning, telling them to beam down:
SPOCK: I have their surgeon's log, Captain. Their last log entry, Captain, on screen.
(A man in a blue shirt is struggling to try and sit up in the Captains chair and dictate the log.)
DOCTOR [on viewscreen]: If you've come aboard this ship, you're dead men. Don't go back to your own ship. You have one chance. Get down there. Get down there fast.
Did the disease really crystalize everyone so quickly that it prevented any of Exeter's crew from beaming down?