Why did this happen to Eden in the episode "Postpartum" (season two)?

Because the society does not operate with a logical or practical approach to their fertility problem to begin with. It operates on male patriarchy & Religious piety through grand practices of symbolism and is governed through fear.

Margaret Atwood New York Times 2017 Interview:

The second question that comes up frequently: Is “The Handmaid’s Tale” antireligion? Again, it depends what you may mean by that. True, a group of authoritarian men seize control and attempt to restore an extreme version of the patriarchy, in which women (like 19th-century American slaves) are forbidden to read. Further, they can’t control money or have jobs outside the home, unlike some women in the Bible. The regime uses biblical symbols, as any authoritarian regime taking over America doubtless would: They wouldn’t be Communists or Muslims.

The modesty costumes worn by the women of Gilead are derived from Western religious iconography — the Wives wear the blue of purity, from the Virgin Mary; the Handmaids wear red, from the blood of parturition, but also from Mary Magdalene. Also, red is easier to see if you happen to be fleeing. The wives of men lower in the social scale are called Econowives, and wear stripes. I must confess that the face-hiding bonnets came not only from mid-Victorian costume and from nuns, but from the Old Dutch Cleanser package of the 1940s, which showed a woman with her face hidden, and which frightened me as a child. Many totalitarianisms have used clothing, both forbidden and enforced, to identify and control people — think of yellow stars and Roman purple — and many have ruled behind a religious front. It makes the creation of heretics that much easier.

The whole idea of handmaid itself wouldn't be so bad if this was someone's choice, but it is not. These are women of the former United States whom are seen as brooding stock. Their lives before are not to matter and they are to have nothing of their own, let alone the absurd situation of sexual practice (rape) that happens, as there could be other ways to achieve fertility without this ritual, including artificial insemination. Season two also revealed through Canada scenes that the rest of world may have begun to solve their problem with science, as Mrs. Waterford is offered a way out.

When one considers the Religious aspect and considering where "bad" Handmaid's go (the colonies) or other things that can be done to them (physical mutilation: surgery, stoning), then one may speculate that the idea of handmaids being "precious" is half-baked (due to psychological and physical trauma) and possibly the notions of the handmaid was meant to be temporary solution and that is where Eden comes in.

Eden was an example of what Gilead has "hoped" will be its future, in which young "pure" women will be married and will be able to have their own children and remain faithful to their husbands no matter what, but Eden interprets the teachings differently...

Confirmation and Additional Storytelling Explanations from the Eps

The episode features the death of a major character: Eden. Was her death always part of the design of the character, or was it something that came up organically over the course of developing the season?

I'm not sure if we knew as soon as we conceived Eden that she would not last for the season, but pretty early on in the discussion phases, we made that decision. And I think it's a really impactful death. First of all, it's a character that seems to be really polarizing for the audience. A lot of people assumed that she was going to be the big troublemaker, the person who was going to stir up problems for June and for Nick. She seemed like such a pious and true believer, and as it turns out, it's those very qualities that get Eden in trouble. And she does have a very big impact on everybody, except not in the way that we expect.

I find it heartbreaking that here we have this pure, innocent soul who grew up drinking the Kool-Aid that Gilead's been serving, and she just wants to live her life in the way Gilead taught her. She wants to be true to herself and to God, and in her mind, God knows what's in her heart. So she can't deny this love she thinks she found with Isaac, and she needs forgiveness from Nick for the disloyalty that she has shown him, but she's sticking to her beliefs. She's really one of the most honest people in this whole community. I find her death obviously really tragic, but it's horribly beautiful in a way, too.

Adultery (against a man) is something that we begin to learn is taken very seriously, as Mrs. Waterford found out what happens when she oversteps her husband, despite her own grand stature. So Eden's fate, although impractical, makes philosophical sense for what Gilead perceives as Godly, since babies are now being born. Now whether they are currently being too hasty and things might backfire on them remains to be seen and is a possibility, but this is about privileged male ego and dominance in form of a authoritarian government through religious extremism.