Plato's own argument against Protagoras's relativism brings out this point. It is that relativism itself uses the very principle of absolute truth that it explicitly rejects. Relativism cannot even make its own claim without holding that it is above the relativity that it postulates. But if relativism is not absolute, then it allows its opposite, namely absolutism. Relativism could be true only if it were relatively relative, and that is not a denial of absolutes.
Subjectivism has the same problem.
If there is no objective truth, that would be an objective truth. So if subjectivism were true, clearly we couldn't know it, we would just have our subjective impression that no one would need to pay any attention to. As you have seen, logic requires premises, and it ultimately cannot prove those premises. In Plato's day logic didn't exist yet, so it wasn't until Aristotle that the regress of reasons could even be described.