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This came up on another question, where in A Song of Ice and Fire (the books Game of Thrones is based on), the Unsullied have all of their genitalia removed, not just the penis.

"In Yunkai and Meereen, eunuchs are often made by removing a boy's testicles, but leaving the penis. Such a creature is infertile, yet often still capable of erection. Only trouble can come of this. We remove the penis as well, leaving nothing. The Unsullied are the purest creatures on the earth." He gave Dany and Arstan another of his broad white smiles. "I have heard that in the Sunset Kingdoms men take solemn vows to keep chaste and father no children, but live only for their duty. Is it not so?"

_{-- A Storm of Swords, chapter Daenerys II}

Removal of the penis makes sense in such a context, both to force obedience and to degrade the soldiers, but if you castrate a man then he loses most/all(?) of his ability to produce testosterone. Testosterone, being desirable for a soldier that relies on strength and fighting prowess, would be foolish to get rid of.

So for a (supposedly) elite army, why would it make sense to do so?

The unsullied are an analog of the Roman Legion.

The most important quality in a legionnaire wasn't his testosterone fueled aggression, it was discipline.

In fact - when holding a shield wall - the last thing you wanted was a hothead beside you breaking formation and bringing the attack to the enemy. You held, the guys behind you pistoned their spears forwards between your shields, your did up close work with your Gladius (short sword).

Fully castrating the unsullied removed a major distraction, their upbringing/training forged them into perfect Legionnaires. They ignore wounds, hold the shield wall, and slowly and carefully wear down their attackers. They were made almost into automata, no humanity, no distractions. They would hold or advance as commanded, regardless of what was thrown at them.

Their lack of testosterone worked in their masters favor.

The existing answers already cover the question, I will just add a quote, from the same chapter as yours, were Daenerys understood that all soldiers are eunuch, and ask the same question to Kraznys mo Nakloz

Some of the soldiers were tall and some were short. They ranged in age from fourteen to twenty, she judged. Their cheeks were smooth, and their eyes all the same, be they black or brown or blue or grey or amber. They are like one man, Dany thought, until she remembered that they were no men at all. The Unsullied were eunuchs, every one of them. "Why do you cut them?" she asked Kraznys through the slave girl. "Whole men are stronger than eunuchs, I have always heard."

"A eunuch who is cut young will never have the brute strength of one of your Westerosi knights, this is true," said Kraznys mo Nakloz when the question was put to him. "A bull is strong as well, but bulls die every day in the fighting pits. A girl of nine killed one not three days past in Jothiel's Pit. The Unsullied have something better than strength, tell her. They have discipline. We fight in the fashion of the Old Empire, yes. They are the lockstep legions of Old Ghis come again, absolutely obedient, absolutely loyal, and utterly without fear."
_{-- A Storm of Swords, chapter Daenerys II}

Soldiers do not need to be strong to fight.
Soldiers do not need to be skilled at fighting to fight well.
All a soldier needs to be able to do is obey orders without question or hesitation. And most importantly to never break formation or retreat from the fight.

Battles--especially during the middle ages--are fought until one side loses the will to fight. War is not about killing your enemy it is about convincing your enemy to give up and stop resisting. They are about breaking your enemies will to fight, so that they become subservient to you.

The unsullied are an elite army because they are disciplined and do not fear death or pain. They will continue on fighting when other soldiers would have run in terror for their lives.

It is worth mentioning that the Unsullied were never really meant to operate as a whole elite army.

They were known as the best infantry in the world, but mostly sold as slaves perhaps in batches of 10 to 100. Not necessarily just to regents intending to use them in war, but to also to merchants and similar, who wished to use them in times of peace.

Clients who perhaps wished to avoid having soldiers constantly in search for either money, booze or women. And so their discipline outside combat might be just as important as their discipline in combat. Idle, undisciplined soldiers means trouble.