Plato is the one figure in this company whose primary works survive complete and whose historical existence is beyond question, which makes him a useful control for reading everyone else.
His importance for the esoteric tradition rests on a handful of texts read in a particular way. The Timaeus presents the cosmos as a living being with a soul, constructed by a craftsman god on mathematical proportions, with the four elements assigned to four of the five regular solids. The Republic ends with a myth of judgement and rebirth; the Phaedrus and Symposium describe the soul's ascent through beauty; the Phaedo argues its immortality. Taken together they supply almost everything a Western mystical system needs: an ensouled and ordered universe, a soul that predates and outlasts the body, and a graded ascent from appearance to reality.
The refraction matters more than the source. For most of the tradition's history Plato was read through Plotinus and Proclus, whose systems are considerably more hierarchical and more religious than anything in the dialogues, and in the Renaissance he arrived in Latin through Ficino, who translated him alongside the Hermetic writings and presented both as branches of one ancient theology. The result is that a great deal of what is called Platonic in esoteric writing is Neoplatonic, and some of it is neither, the Timaeus does not teach emanation, and Plato has no doctrine of the One.
The tradition of an unwritten teaching reserved for the Academy is real, Aristotle refers to Plato's lectures on the Good, but what it contained is a live scholarly dispute, not a secret doctrine anyone has recovered.