There is no rule saying they will automatically see you, you made that up.
In combat, they will 'usually' see you if you leave your hiding spot. But you are not 'usually' trying to be sneaky against someone in a different room that may or may not even be aware you are there. This situation is, by definition, unusual.
Let's see what it says: "In combat, most creatures stay alert for signs of danger all around, so if you come out of hiding and approach a creature, it usually sees you. However, under certain circumstances, the Dungeon Master might allow you to stay hidden as you approach a creature that is distracted, allowing you to gain advantage on an attack before you are seen."
So, if you come out of hiding, you are "usually seen". What is the circumstance in which you AREN'T seen then? Luckily, we don't have to guess, it says so in the next sentence. The DM can rule that a creature might be distracted enough to allow you to stay hidden.
Which, when we combine it together says "When you leave hiding all creatures see you unless the DM rules they are distracted."
I've ruled that creatures are distracted when they were intent on looking another direction, when they were talking to someone, while they were in the middle of playing card games, and so on. Thereby allowing you, outside of combat of course, to sneak past people without needing cover.
First, there is *no* rule saying you are 'automatically detected' if you do not hide. That is absurd.
Why is that absurd? I'm with the other people who replied. If someone is standing in front of you making no attempt to hide, they can be seen. Otherwise the world would be a weird place with people randomly vanishing. Eyesight could not be trusted.
This doesn't need to be a rule. The authors of the rulebook felt there were certain things common sense enough not to have to publish in the book. One of them is that people can see normally if nothing is changing the situation.
Second, the Hide action is one way to be undetected, it does not say it is the *only* way.
You are making a lot of assumptions.
The book is a list of things you CAN do, not a list of things you can't do. If the DM wants to let you stealth another way, then they are allowed to do so. However, the only option listed in the book to hide is to use the Hide action. Which takes an action since it is listed in the actions in combat section.
The Dex(Stealth) skill says it can be used to sneak past a guard, slip away, or sneak up.... it *never* says anything about *requiring* the Hide Action.
No, it doesn't. But as other people have said, using a skill in combat generally requires your action unless otherwise specified. Stealth never says it DOESN'T require an action. Plus, there's already a rule that says if you use the Hide action during combat you need to roll your Stealth skill and describes what happens when you do. Why would that action even exist if you could just "Stealth" from people which is apparently better than using the Hide action since it doesn't require an action?
Rule number 1 I use as a DM is to use the tools provided before inventing new ones. The book tells you how to hide from people. It's better to use the rule that has been given before coming up with a new one.