Did these apps have barriers to make it difficult to buy things? Or did the design allow for unrestricted access to these purchasable goods? Is purchasing as easy as pressing one button and being done? Isn't it an issue that developers should account for while designing their apps?

Is this the game developers responsibility though?

iOS devices have settings that can help to control this by disabling or restricting IAP. Google Play offers similar settings for Android devices. Whenever I make a purchase on Android I get an email within minutes (usually almost instantly) with a receipt and although it's been a long time since I've regularly used an iOS device I remember it being similar.

They didn't always have those settings though. They were introduced because some developers were (seemingly) intentionally abusing the lack of Apple/Google/government policing to exploit the situation.

Even after those settings were introduced, some developers continued to look for ways around it. With iOS, after parents inputted the password, there was a 15 minute window where kids could spend as much as they want without parental approval before the password needs to be input again.

That 15 minute window may have been fixed. So perhaps those same developers are looking for other ways to exploit that ignorance.

If this is an issue of children being disobedient, fine - that's on the parent. But some of these games are targeted to really young ages, and how clearly can even older children discern the difference between "Buying" fake stuff with fake "Gold" and "$$$" using in-game currencies like in any RPG ever invented, and buying (fake) stuff with (real) money. Until they get burned (by their parents getting burned, and them getting a talking-to), they won't learn that "Purchase" with one color means fake money, and "Purchase" with another color means real money.

They won't learn, until they are forced to learn by unexpected consequences, that $ means fake money in some games, and real money in other games. They won't recognize that the standardized iOS pop-up window means something special until they learn the difference, most likely the hard way.

I would argue that parents should be policing their children's spending through the use of these settings and appropriate supervision

Absolutely. But not every parent is tech-savvy. Until they get burned one or more times, they may not even realize that children can spend money inside a game, without the parent inputting a credit card number.

You grow up your whole life, and you can spend money in two ways: You had over cash, or you temporarily hand over a credit/debit card (and must confirm with a PIN number).
How is a parent supposed to know that after buying one thing, like a game for their kid, the kid (unknowingly) can then go and spend additional money without further input from the parent?

It's not unethical for in-app purchases to exist. It's not unethical for game developers to use them. There's a partial burden of responsibility on platforms (Android/iOS and even online shops Amazon or iTunes) to provide 'reasonable' safeguards/tools and make pop-up messages clear. If a corporation fails at that, they probably aren't being unethical, just negligent.

But it is unethical if some game developers intentionally try to exploit parent and child ignorance of the technology to trick children to spend their parents' money without permission.

Now, proving a developer operated out of maliciousness can be hard. However, since we're merely discussing ethics and not solutions, I think this is a developer-related ethics issue (if any developer tries to exploit it). It's also a parental ignorance of technology issue.

Everything "not against the law" isn't automatically ethical. Ethics is not adherence to the law - the law is a (clumsy) attempt to codify ethics. But the law does it in a general case, and doesn't cover every circumstance, and trips over itself in multiple areas. "You didn't forbid me from stabbing him, just shooting him. So technically I didn't do anything wrong!". Spirit of the law vs letter of the law. Except, you want don't actually want to focus on the spirit of the law either (spirit = what the judge/lawmaker "meant" when he originally wrote the law), you want to look at what the law was meant to reflect: general morality. So if the law fails (in law and even in the 'spirit of the law') in some area to predict or cover one area of morality, that doesn't excuse people from not being moral.

...while designers should be providing the best possible experience for their players -- making purchases as simple as possible is part of a good experience.

Agreed - technology itself isn't the problem.

In my opinion, ignorance of technology is part of the problem (this will probably resolve itself fairly rapidly - people will be forced to learn it, and they will). But intentionally exploiting that ignorance to rob someone is an example of developer-related ethics.