Not always . . . it’s not a given. See HIV, hepatitis, several strains of flu, West Nile, Ebola, and Zika.
One of many articles pointing out that “the model” isn’t universally as you assert. . https://apnews.com/article/fact-checking-011488089270
A literature search would lead to tons of articles the hows and whys tip-off viruses evolving to be more or less lethal to their hosts (both human and not human).