To my mind, there are several reasons why this isn't the standard procedure. One reason is that Likert-type items are ordinal data, which means that standard statistical methods such as linear regression and t-tests shouldn't be used. Instead, non-parametric tests need to be used, and those generally confer lower statistical power. In contrast, a scale score (the sum of several Likert-type items) is per convention handled as interval data, which means that those standard methods can be used.
Another reason is that this might increase the probability of false positives. If a scale used to measure depression has 17 items and the significance level is set to p < 0.05, there is a 1 - 0.95^17 = 58 percent chance of at least one false "significant" association between the test group (treatment vs. no treatment) and an item from the scale.
Furthermore, the idea (which is questionable) is that there exists an underlying pathological "depression process" in the brain, which is expressed through different symptoms such as depressed mood, sleep disturbances, loss of interest etc. This is the medical model, and it is analogous to, for instance, pneumonia, in which there is a pathological infectious/inflammatory process in the body caused by bacteria, and giving rise to symptoms such as coughing, fever, etc. From this theoretical point of view, it makes more sense to assess the total score of the depression scales, which is thought to represent the "latent variable" that is the "depression process", rather than just one of the isolated symptoms.
However, there are examples in the literature of the approach you suggest. Below is one example of a meta-analysis of some trials of SSRI for depression. The study is not without flaws in my opinion, so the results should be interpreted with care, but it is an example of the idea you had.
Influence of baseline severity on the effects of SSRIs in depression: an item-based, patient-level post-hoc analysis
