If you don’t subtract the log complimentary CDF for a half Student t distribution (or another distribution defined over the entire real line), then the log kernel of the posterior distribution would be off by a constant and some post-estimation methods, such as Bayes Factors, would be incorrectly calculated. Although doing so does not affect the posterior draws, packages such as brms have to be strictly correct because we cannot know what the user is going to do with the results subsequently.