I've often heard people refer to Barth as a more "ad hoc" rather than systematic thinker, but I think this is actually a result of misreading Barth's style - particularly his use of repetition. Barth is actually just as systematic and philosophically rigorous as Aquinas. In the first volume of the Church Dogmatics, Barth summarizes a short section on systematic presentations of doctrine with the words "fear of scholasticism is the mark of a false prophet." Barth says, following a similar interest in Protestant Scholasticism, that Preachers need an organized, philosophically rigorous dogmatics in order to not contradict themselves. (Part of the way I'm reading Barth currently is to look at his reception of Protestant Scholasticism, an aspect of Barth's thought that John Webster points to.) This is different from saying that we need our dogmatics to answer every question posed by modern science or philosophy.
Yet at the same time you hear this language of "sin" or "unfaithfulness" sometimes used for systematic presentations of doctrine like in Paul Tillich or in "Neo-Scholastic" Catholic theology. You often hear that theologians writing systematic presentations of doctrine lose the mystery or impose alien categories on the Gospel.
Thinking through Christian teaching in a systematic way is good when it is used to excite the intellect towards the contemplation of God. Aquinas receives from Augustine and the Greek Fathers a strong notion of theology as a participation in God's own triune life through contemplation. Yet when one writes or teaches systematic theology not trying to seek to know God's will and to love God more deeply, but for the end of human intellectual satisfaction, then perhaps one has a problem. One of my professors at Duke, Willie Jennings, said that its important to always return to the fact that we are creatures when we are doing theology. He said that acknowledging that we are creatures frees our desire to know from the fear of not knowing or the obsession of knowing completely.