The Athenians followed some sound procedures in their legal cases that we respect today, like having the jury decide one's guilt or innocence and letting the defense speak last. We know a fair amount about how courtroom speeches worked, not to be sure in Socrates' own time but in the decades after his death, because we have over 100 speeches by plaintiffs and defendants.
Plato tells us a lot about the trial of Socrates but we can't trust him entirely. After all his version differs from that of Xenophon. So anything we say will have to be provisional and attentive to the limitations on our evidence.
Some of the weaknesses of ancient trials followed from the difference in technology between then and now, which translated into a difference in credentials and proof. John Dillon's book SALT AND OLIVES spends a lot of time discussing ancient legal cases, and sometimes points out how things we consider trivially easy had to be argued in court then. For instance, an ID card and birth certificate today usually proves that you are the child of the deceased. Lacking those documents, the Athenians would make roundabout arguments. ("No one stopped me from going to the family funeral, so they obviously treated me as related to the deceased," etc.)
But from our point of view, the most glaring procedural difference between then and now is the absence of a judge. Each side made its case, and a jury decided (a large jury too, 501 in a capital case); but no one oversaw the proceedings. No one was there to declare some evidence inadmissible, or to distinguish between evidence brought forward and evidence relevant to a charge. So people could bring in all the accusations they wanted to, as long as the jury was receptive. (Aristophanes' comedy WASPS depicts jurors as vengeful old men. Venal, too: If a demagogue wanted them to vote his way, he'd threaten to end jury pay.)
A judge might have made Socrates shut up, when he wandered into talk of the gossip that had been circulating about him. But it might also have stopped his accusers from prosecuting a case as vague as theirs.
So why didn't the Greeks think about putting a judge in the courtroom? Because they had just finished taking the judges out of the courtroom. Before the full development of the Athenian democracy, many legal cases were heard by panels of judges that later went before a jury. The Athenians moved from judges to juries in order to make justice more democratic. In the process they lost some of the niceties of a tightly governed judicial system.