Indeed as you and another commenter pointed out correctly, it is not the whole picture, and given an intelligent, evil design one could possibly create a disease that spreads fast and kills only much later (by let’s say, causing cancer). But fortunately we are not inside Plague Inc.
But if I am not mistaken, from a strictly evolutionary perspective there is no pressure towards such a disease — a genetic variation that is more deadly won’t dominate over its less deadly variant and thus deadliness could be a variable that only changes randomly from the initial level (and decreasing if it is actively detrimental to spreading, where I would say that in case of a human population deadliness does correspond with fear).
But IANAE, nor a mathematician.
(An interesting question to chew on is: what was the impact of religious sex taboos in pre-modern and early modern era? If monogamous couples were more likely to have healthy offspring and nurse their kids into adulthood, it might explain some of the societal swings towards puritanism in the 16th-19th century, which was a period of endemic syphilis.)