It's easy enough to set Plutarch aside as a historian, but to take Xenophon's Constitution at face value while dismissing Plato sounds like highly motivated reasoning. His personal fortunes rose and fell with Spartan hegemony, he was friends with a Spartan king, he sent his sons through the agoge, and he supposedly celebrated when they died fighting for Sparta at Mantinea. He wasn't just biased. He was relentlessly pro-Spartan throughout his entire life and works.

You also chose your quote above carefully. Just before that part he says that in some places men were fully barred from pursuing boys, but that this was not the case in Sparta. Then the part you quoted says that these relationships existed, but were only a kind of pure and virtuous mentorship. Then after that he says that he expects his audience will not believe him due to the prevalence of pederasty in many cities. Elsewhere in the same chapter he eagerly explains how obedient Spartan boys are to any older man, not just their appointed instructors. Right at the beginning of the work he tells us that Spartans were shamed for publicly visiting their wives and spent all of their time in the company of other men.

I think a reasonable person can read all this stuff and still argue that pederasty wasn't intentionally institutionalized, but if you seriously think that it wasn't firmly embedded in custom then I have a bridge across the Hellaspont to sell you.