Socrates is actually right in the last clause, because neither the ideas nor the souls existed before birth, partially because birth is an arbitrary limit.
The use of birth as a delineation is entirely arbitrary and is rooted in the same kind of inaccurate conception of identity and consciousness that underpins Socrates' entire worldview. The prenatal knowledge Socrates imagines he has observed exists before birth in that it is encoded into a human's DNA well before any given baby passes through a birth canal, but there is no evidence for that baby somehow being filled with knowledge or consciousness at a certain point such that one can talk about before birth and after birth as useful time designations. Again, Socrates' argumentative and logical failures are largely born out of scientific ignorance, but this does not lessen the fact that he is not so much making a genuine argument as much as making things up.
However, what makes Socrates so effective is that after he makes up his initial assumptions (regarding the existence of a soul and gods, for example), he attempts to discuss those made-up things in straightforward, logically cohesive ways, so that his explanation for the eternal nature of the soul appears reasonable when compared to the responses of the straw men around him. In other words, Socrates appears convincing in Phaedo because he knows the rules of his imaginary mythos better than the other characters, and thus can outline those rules in a way that has the appearance of critical investigation and logical progression. Thus, his conversation partners end up reiterating Socrates' most problematic assumptions, such as when Simmias says "there is nothing which to my mind is so patent as that beauty, goodness, and the other notions of which you were just now speaking, have a most real and absolute existence" (Plato 46).
These things do have a real and absolute existence, but only as the meaning-content of human consciousness, and not as anything that transcends the limits of physics. While there may be some evidence to suggest the existence of something metaphysical, all evidence presented throughout human history thus far has failed under investigation, and so to have statements like this reiterated unchallenged as a part of Phaedo's message only serves to paper over the very real gaps in logic. In essence, Simmias' reiteration of the eternal, metaphysical existence of ideas as an apart from physical existence is one of those instances where a magical statement is made without evidence and is simply taken as true. Thus, the argument once again returns to Socrates' fundamental assumptions regarding the nature of the universe, because his entire discussion of the soul is framed within the hard limits of these fallacious assumptions. Socrates simply cannot escape from his ignorant belief in the soul, and thus it is only natural that he should be able to also find evidence for its supposed immortality. His position regarding the soul is no different from someone who simply states that a magical butterfly controls and motivates everything, and then immediately starts seeing evidence of the divine butterfly's handiwork in everything, which then serves as evidence for the assumption regarding the butterfly's initial existence; the only difference is that "soul" is an older and more socially acceptable magical concept than a divine butterfly.
Tracking the argument in Plato's Phaedo helps to demonstrate how discussions of the soul do not have any real basis in reality, but must always begin from a fallacious assumption rooted in magical thinking. Socrates' discussion of the soul's eternal nature and his attempt to prove this nature though his recollection theory depend first on the faulty assumption that a soul exists in the first place, and because Socrates is unable to provide evidence for this claim, his arguments for why this imaginary concept might be also eternal are almost irrelevant. However, it is still useful to examine them in order to see how the appearance of logic can be used to make an argument rooted in faulty assumptions appear to have the legitimacy of reasoned criticism.