The video is misleadingly entitled "Prof Alvin Plantinga on Reasons for God," because he doesn't really give any reasons, let alone argument. I think it's perfectly true, as he often avers, that the theist is within his epistemic rights to believe in God even in the absence of rational arguments, just as we often find ourselves reasonably believing all sorts of things we cannot prove, such as the reliability of our memories, sense experience, self-perception, being awake rather than dreaming, and even such curious things as the falseness of Bertrand Russell's hypothetical proposition that the world popped into existence five minutes ago with all the appearance it has had since then of great antiquity. But it's not a demonstrative argument, as much as it is a reasonable testament to common epistemic experience.
St. Thomas Aquinas himself says in his Summa Theologiae, Q. II, Art. 2, ad 1:
The existence of God and other like truths about God, which can be known by natural reason, are not articles of faith, but are preambles to the articles; for faith presupposes natural knowledge, even as grace presupposes nature, and perfection supposes something that can be perfected. Nevertheless, there is nothing to prevent a man, who cannot grasp a proof, accepting, as a matter of faith, something which in itself is capable of being scientifically known and demonstrated.Nope, nothing wrong with simply believing in God because one finds himself believing in God; and this needn't be seen as a form of fideism or "blind believe-ism" provided one does not close the door to reasoning about it.