It is a studied phenomena that languages effects how we see the world, problem solve, how we act. They did take it to a bit of an extreme, but it's still logical with some filler.
All language is is concepts that can be shared with another. I say "Red apple" you know what I mean.
At first, Louise must understand the basics of the language. once she does, she can read the rest of their language (that final message they sent), which could very well include concepts on time "travel".
Outside of that, the structure of a language has implication on how one sees the world.
Being that the alien language is in loops, we can assume that in base their language is like a programming language, and each "word" is really a function. Once you understand a function, you can predict how it'll pan out over time. If you understand the world as functions or processes, you can likely predict the outcome of something. Much like plugging in variables in a math problem.
We have to make the assumption that each of those alien "words" stacked together create a very elegant predictive pattern on how to see the world. and once you understand that pattern, you are able to see the circular looping of life.
here is where it breaks down. it has to be an incredible complex or elegant language to handle every single variable possible. So either they sped up the time she needed to fully grasp it, or it's making a statement that there is an underlining principal of nature we do not see or grasp, that this unlocks. (the latter may be more likely, considering the movies statements about zero emissions from the space craft)

Thats not a flaw.
Its what we call a "premise". As in what if..
You literally point to the main focus point of the entire script, what if we could see our future, would we live differently, and say its a flaw since this isnt the case?
Nothing we do can change this.
You are no authority of advanced physics, so dont talk in absolutes like you know the inner workings of the universe and what is and isnt possible.

What if she didnt travel in time per se but rather only experienced the future time. Theres the idea that everything - the universe - is a data set and with enough information/data you can know everything, right? Sort of like how when we know the gravity of a planet and the weight of a ball (among other variables) we can know how long it will take fpr that ball to reach the ground when its dropped from a certain height - in this sense we essentially can predict/already know the future, and once tht ball is dropped the future is already in motion and is inevtiable. What is in motion from the "beginning" of the universe is already contained in the universe and will always play out.
So what if this new language just helped us tap into the data set and Amy Adams got to see that data already contained in her, got to see what motion will play out, got to see HER future essentially. All her memories/fladhforwards were only from HER perspective because thats the data she has available in HER brain.
Where it gets confusing is the nature pf exactly what she tapped into, because it may have only been future timeline #1, and there could be many timelines that are available or ready to change. For example I know her character decides to have the child like she saw in her flashes, but unless she plays out literally evrrythih the same as she saw(exact same movements, pauses, sentences etc) then shes not actully living in the future she had witnessed in her flashes but rathrper in a new future that incorporates her knowledge of her having the flashes.
Ps not sure if this makes any sense

Einstein's equations allow for backwards time travel.

Well, if we're going to get picky, Einstein has no opinions, he's dead, it's what happens in linear time, people die. Either way your argument is invalid, just because it's science fiction dosen't mean it can ignore it's own rules, and this is set in our world in our time so they have to follow those rules. Notice how non of us have a problem with 12 spaceships landing, that didn't really happen either but we accept that. And it just being a story does not make it exempt from being disliked.
Besides, I'm pretty sure Einstein never said anything about time being non-linear, the theory of relativity only shows that time doesn't move at the same speed under all circumstances, it never says it can move any other directions than forward and it never says anything can exist in anything but the present. As far as I understand it anyway, not that I know that much about it.

Einstein (and other eminent physicists) have a different opinion. And, this is science fiction the idea extends the idea that a different language affects how you think, then a sufficiently different language can affect how you perceive. You might wish to think it nonsense, but it's just a story

By your logic, any film that defies the rules physics should have a plausible explanation as to how? Science FICTION!!! I'm with you though, I'm still disgusted that I never got to see exactly how Luke built his light sabre. Ruined the entire franchise for me.

Dr. Banks is to old to get pregnant and carry to term successfully?

The "flavor" at the end of 2001 is diffrent, so is Contact and even Interstellar. Contact does a great job of conveying a concept we just couldn't understand. This movie however is not using a metaphor, it's actually time travle, the entire movie depends on it, it's stated clear as day that that is what is happening, and I for one don't like having to make up excuses why a movie should work when it's clearly not in the movie. That's like me saying that James Bond appearing as diffrent people through out the series is because he's actually Doctor Who.

It's just a metaphor for the aliens giving humans a transcendent truth. Obviously the film itself cannot be expected to contain such a truth. So the whole time thing is a flavor of that, like the flavor at the end of 2001: A Space Odyssey. Don't take it literally.

I liked the concept. I do not think I have time figured out, nor do you. Time may be pure perception, and there may be a language that taps into our brains and has the ability to rewire our brains to be able to perceive the future as the present because time may not exist. What if we arent as smart as we think we are, but are just above a mosquito relative to how intelligent we could become? When we consider this, the limitations you put on time seem to simply get in the way of the scenario in the movie.

I am on your side, I found this movie annoyingly dumb.

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So they literally broke physics with no corresponding theory to replace time to make the plot work.. that's quite amazing, and is too much of a stretch for me. Even fiction has rules they need to follow. If you get rid of a basic tenet of physics, at least try and replace it with a half decent alternate theory for the story to continue to make sense!

The argument they have is that time actually doesn't exist and that she just learns to let go of the concept through the language. That only makes sense if this is a simulation though and it becomes like the matrix. They say we couldn't comprehend how it works which is just a cop out.

The perception and passage of time transcends every human construct, and is experienced equally by all things living / non living. The basic premise of the movie that changing the language will somehow give us the properties of faster than light photons (as an example) which may perceive time differently than all other matter, is ridiculous, but could perhaps fool most of the general public.

"But the fact is, everything, our entire lives, even our languages and our brain and how we percieve and relate to time are all shaped and built and controlled by time's properties and time related effects such as causality and the arrow of time."
I think the counter-argument, within the context of the movie at least, is that this perception of time has been shaped by language. In other words, series of Subject>verb>object relations all in sequence creates a limitation in our perception.

Yes, but the premiseitselfhas plot armor. And you don't. So prepare yourself for rebuttals galore and other general nastiness from the fans.
Enjoy these words, for one day they'll be gone All of them.