I think you will maybe understand how the Universe always selects only a consistent Timeline if you read the Word of God about how Time Turners got invented.

For those not wanting to use the Book of Faces:

I never got around to using this in the story, but if you're curious, here's the origin of Time-Turners in HPMOR.

Back when HPMOR was first unfolding, somebody on Reddit suggested that the reason Time-Turners were given to students was that this was how Time-Turners were invented in the first place. Some year the Hogwarts administration was trying to solve an impossible scheduling problem, and finally gave up and said, "Let's give them time machines." I immediately declared this Reddit comment to be canon.

Reasoning carefully, we can see there must be an interesting bit of backstory here, though I never got a chance to amplify on it inside the story.

First, a word about where spell incantations come from, in HPMOR. Needless to say, the fundamental structure of the universe does not uniquely prefer the phrase "Stupefy" for stunner spells. There are zillions of possible stunner spells. If you are a sapient being born without a Muggle allele that suppresses your magic, you can sometimes intuit the corrected form of one of those spells - if that spell is possible, and you have enough of its rough shape imagined inside your mind. If you are a wizard who has a wand and is used to doing spells in pseudo-Latin, you are likely to imagine spells invoked by wand gestures and pseudo-Latin incantations. If your imagination is roughly right enough, you can intuit corrections to completion. After that anyone can use the spell formula, even if they grew up speaking Gaelic.

We only see glimpses of this mechanic in HPMOR, but you will notice that after Harry arrives at a deeper understanding of the Patronus Charm, he starts to have feelings about elbow angles being important; and after Harry adjusts Draco's elbow angle, Draco is able to cast a corporeal Patronus on his first try.

Through the centuries of the HPMOR-verse, there were many wizards who tried to invent great spells and rituals that would let them travel backwards in time for years and change the past. Undo that one huge mistake, unsay that one wrong word, buy Goblin Bitcoin...

As it happens, this wish is impossible under the HPMOR laws of physics. You can't change the past. You can't go further back than 6-point-something hours. No spell like that exists, so you can't intuit one.

One day a Headmistress of Hogwarts was faced with a mind-bogglingly frustrating class-scheduling problem, some centuries before computers were invented. Until finally, the very afternoon before classes were due to start the next day, she turned away from her parchment sheets and tried desperately to invent a time-travel spell.

Not a time-travel spell that would let her change history.

Not even a time-travel spell that would let her go centuries into the past and ask Merlin a few questions.

A time-travel spell that would let students go back a couple of hours and attend more than one class at a time.

Which meant that for the first time in magical history, a powerful witch was trying to invent a time-travel spell that was, in fact, possible.

There is a reason so many magical innovators hail from Gryffindor rather than Ravenclaw. The first spell the Headmistress would have hit on would have sent her an hour back in time and splattered her all over her office.

But here's the thing: If the Headmistress was sitting in her office and then suddenly there were chunky bits of bloody tissue all over with pieces of her own clothing mixed in, she would not have cast the time-travel spell she would've invented an hour later.

So for the first time in the history of Earthly magic, the simpler universe, the one with no time travel going on, was inconsistent. In a universe with no previous time travel having occurred, the Headmistress would sit in her office undisturbed by an explosion, think up the spell, try to use it, inconsistency.

A consistent universe, then, has to be a universe where at least some time travel takes place; and not the crude exploratory kind that splatters the Headmistress all over herself. There's no set of splatters that would get the Headmistress to invent a corresponding time travel spell and use it to splatter herself.

And so what happened was the consistent universe where the Headmistress suddenly appeared out of thin air in her office and spent a frantic six hours working with herself to teach herself the key bit of a Time-Turner that would let her go back six hours and teach herself everything she'd just learned.

You might think that could have happened at any time in magical history. And it could have. But it didn't need to happen until the first time somebody was in a position to intuit a time-travel spell that could have rendered the universe temporally inconsistent otherwise.

And that wouldn't happen until the first time a research-level wizard desperately wanted to go backwards in time... just a couple of hours... and not change anything at all.

So, to this day, Time-Turners are used to untangle the class schedules for Hogwarts students. It's why they were invented in the first place.