Unsure asked:

Did Plato propose in the Republic that all people are born with equal souls so they have an equal
chance of ruling OR men and women with the correct mix of metal in their soul can rule? I tend to
think it is the correct mix of metal.

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Let's see what Plato has to say here:

"'We shall,' [Socrates] said, 'tell our citizens the following tale:

"You are, all of you in this community, brothers. But when god fashioned you, he added gold in the
composition of those of you who are qualified to be Rulers (which is why their prestige is greatest); he
put silver in the Auxiliaries, and iron and bronze in the farmers and other workers. Now since you are
all of the same stock, though your children will commonly resemble their parents, occasionally a silver
child will be born of golden parents, or a golden child of silver parents, and so on. Therefore the first
and most important of god's commandments to the Rulers is that in the exercise of their function as
Guardians their principal care must be to watch the mixture of metals in the characters of their
children. If one of their own children has traces of bronze or iron in its make-up, they must harden
their hearts, assign it its proper value, and degrade it to the ranks of the industrial and agricultural
class where it properly belongs: similarly, if a child of this class is born with gold or silver in its nature,
they will promote it appropriately to be a Guardian or an Auxiliary. And this they must do because
there is a prophecy that the State will be ruined when it has Guardians of silver or bronze."

Republic415a-c, Lee translation

So you suspect correctly: Plato believes that different people are born with differently composed
souls, and only those with the right sort of soul can become rulers. In modern parlance, he believes in
equality of opportunity, but not in equality of outcome.

But this passage should not be taken out of context. The Republicis a work of great literary
sophistication: at the very least, Socrates's tale is a story within a story. He is relating a myth (story 1)
which is not supposed to be taken as literally or even metaphorically true, but rather as a pragmatic
device to persuade the hypothetical citizens of the ideal republic which he is describing to Glaucon
(story 2) to follow the rules laid down for their benefit. And even story 2 is not just an exercise in
political speculation, for Plato also maintains a running analogy between the constitution of the state
and that of the individual: everything that he has to say about the one is intended to throw light on the
other and vice versa.

Which is a roundabout way of stressing that Plato doesn't really think that souls contain mixtures of
metals, but only that individuals have innate capabilities and limitations which disqualify all but a few
of them for rulership.

Andrew Aberdein