Yes, but most of these diseases can be reversed in minutes with a hypo-spray. No such luck when you got beriberi in the middle of the ocean and it's the 16th century.
Not necessarily. Common diseases, sure, but the rare exotic ones were still a threat in Kirk's time. I mean, look at "Requiem" again. That was a life-threatening medical emergency to the entire ship, not a minor nuisance to be cured with a few quick shots.
And, especially in the third season, the Enterprise was often dispatched to deliver vital medicines to some remote colony that was suffering from some ghastly epidemic or another, as well as exotic fungi that wiped out entire food supplies, etc. Plagues and famine seemed practically commonplace.
The Final Frontier was never supposed to this safe, orderly, well-civilized place where everything was under control. Maybe back on Earth, sure, but out on the borderland, in unexplored territory. . . the
Enterprise was not exactly on a pleasure cruise to friendly ports of call.