Using in-game tools/interface (not mods or cheats), what are some ways to manage anger in Sims? Trying to play with how the game was designed and I guess I just don't get this part because Sims get overwhelmingly angry too easily.

Spam rooms with mood objects. Base game has plenty. Such as a wall poster, place it, activate it. Buy another, activate it, then place it right on top of the first so it still looks like only one poster. Rinse and repeat till rooms are so strong with mood changers that no amount of getting angry will matter. As long as the mood objects combined mood outweighs the level of anger (or any other unwanted emotion.) then that will be the main mood of the sim - in that room anyway.

You say no cheats, but the in-game Maxis cheat to shrink objects can also help here to shrink and hide spammed mood objects under other furnishings, if you cannot stand looking at the same decorative mood object in almost every room of the house. The sim family will need plenty of disposable cash, though, to spam the objects without bankrupting the family.

Also there are some nifty Reward Store traits to buy, such as Carefree and Laidback (comes with Island Living.) These two traits practically do the same thing, almost identical in what they do I think, and combined are very potent at warding off high anger. To the point the sim will maybe be a bit too Big Lebowski chill for the rest of their mellow-yellow, carefree, laid back days. It will make you start to wonder what illegal pharmaceuticals and narcotics these two chill-pill traits must come with.