Why did the aliens arrive on Earth with such a large head start in Arrival?

There is only one single timeline. Everything Louise saw, comes to pass. That means the alien language reveals an unalterable future. That also means they know that they had to come now, to have the effect they wanted in 3000 years. That result is presumably immutable.

That said, video recording is much more believable than books written 300 years after Christ, and his followers lived and died. He did live and die BTW - that's unquestionable, there are too many records of him, from both his followers, and his enemies. The question comes not from if he existed, but what exactly he was able to do. Walking on water is possible if the water is frozen for example. Couple that, with 300 years of stories told by word of mouth, and wildly different languages with various possible interpretations and translations, and you get exaggerated facts coupled with fiction, with a dash of questionable interpretations of the original meaning.

So, back to Arrival, if you have a video recording, or thousands, of these aliens, it becomes much harder to say "it was fake". The point though, was the aliens wanted to teach us something, so we'd mature with that tool for 3000 years, and then we could help them back. I'm guessing the time elapsed is key, for the tool to be studied, taught and developed among ourselves.

The key plot point is that the heptapods experience time differently. They experience it at the same time. Louise eventually learns this trick.

This means that to the aliens, it doesn't really matter when they will appear. They will always know (and have known) what happens in 3000 years.

However, humans are still time sensitive, and thus the moment of arrival might need to be very precise. If they arrived earlier than let's say 1900, they would have been dealing with humans who would either feared them, or were technologically incapable of making contact and understanding them.

We can't know the future, but what if humanity would not be as receptive in the future? Suppose that in 2030, hostile aliens land, attack the planet, are defeated, and then humanity decides to never make friendly contact with aliens from that point on.

This means that if the heptapods landed after 2030, that they also would have been dealing with humans who are not receptive to communicating with them.

I can't definitively answer this because we don't know what the (in-universe) future holds, but it makes sense that humans were only willing and able to communicate with the heptapods for a limited window of time - thus causing them to arrive on Earth in this time window.

Edit

A second consideration is that the heptapods specifically split up the information in 12 parts. Louise theorizes that this is specifically to force humans to work together. While it's never confirmed that this was the heptapods' intention, that is exactly what seems to happen.

It's not impossible that the heptapods know that it will take humanity a long time of peace (say 3000 years) to advance civilization to a point where they are able to help the heptapods in the way they need to be helped.

Beside the two already good answers here, I want to add an answer focusing on two points not yet mentioned:

Knowing the heptapods language itself is proof enough

When you speak this language fluently, you are able to see the events of all time at once. So did the heptapods and saw events in 3000 years, and very likely Luise and all others knowing the language will see that as well. At the end of the movie she is teaching this language to others. It is not shown in the movie but can be assumed that other people will learn the language as well and that this will probably accelerate the technological progress of humanity. It can also help to have peace (since a lot of people will be aware of consequences of a war). After all, the language is a gift (I can image that this will become a language studied in schools) and the rescue of the heptapods in 3000 years would be clear to anyone speaking the language.

Even if one can see the whole time at all, actions need time to be done

The heptapods say that they need human help in 3000 year. Image, they are now here and say that they need help. Even if we wanted, we cannot help them because our technology is not advanced enough for example. So maybe it takes just a lot of time to prepare for that help or get humanity trained in something which also takes generations. So they could have found out that this is the best time to arrive in order to give humanity enough time.