And if the true reason was Socrates' past friendly relations with the Thirty Tyrants, Alcibiades, etc., Plato still seemed to have had similar beliefs and moved in the same circles, so why was he left alone? Was he just excused on the grounds of his relative youth and obscurity at the time?
Briefly, because Plato (who had just started his literary career at the time of Socrates' death) was still an obscure figure, with no school or students of his own.
Over his decades as the gadfly of Athens, Socrates seems to have accumulated quite a few enemies. According to Diogenes Laertius:
"there were three accusers [of Socrates], Anytus, Lycon and Meletus...Anytus was roused to anger on behalf of the craftsmen and politicians, Lycon on behalf of the rhetoricians, Meletus of the poets, all three of which classes had felt the lash of Socrates" (Lives of the Eminent Philosophers, 2.39)
The charge, famously, was impiety: "Socrates is guilty of refusing to recognize the gods recognized by the state, and of introducing other new divinities. He is also guilty of corrupting the youth." (Diogenes, Lives 2.40)
In the jittery social climate following the restoration of the democracy, Socrates (some of whose pupils, as you note, had been associated with the Thirty Tyrants) was especially vulnerable to these charges. But in the end, of course, it was his cheerfully impudent defense that got him sentenced to death.
Plato was already known to be a close associate of Socrates by the time of trial. In fact Diogenes, reporting a later tradition, notes:
Justus of Tiberias in his book entitled The Wreath says that in the course of the trial Plato mounted the platform and began: "Though I am the youngest, men of Athens, of all who ever rose to address you" – whereupon the judges shouted out, "Get down! Get down!" (2.41)
Yet despite this public stand (if it indeed happened), Plato simply was not prominent enough to be prosecuted. His early dialogues (which may have just begun to circulate among his friends in 399 BCE) were not yet publicly known, and he had no pupils.
Plato was worried enough about the possibility of prosecution to withdraw to Megara immediately after Socrates' trial (Diogenes, Lives 3.6). He returned to Athens, however, a few months later, apparently after the Athenians showed remorse for their execution of Socrates (2.43).