Why are digits written in groups of three?

Nothing. It's just a matter of convenience and also convention. There are systems that use different spacings. As an example, in India, the first group is of length 3 and the subsequent are of length 2. For instance, we'll write 23,25,963 instead of 2,325,963. So, what's stopping you? Nothing really. But, I don't think it's going to affect your number system in any ways.

It's certainly true that this is an instance of "chunking", but I think that writing numerals that way follows the way we name the numbers in the first place. Consider $123,456,789$. Each $3$-digit block is read as a stand-alone three digit number, followed by an appropriate big-number word: "One hundred twenty three.... million," then "four hundred fifty six... thousand," and finally "seven hundred eighty nine."

Thus, the question is really, why did we stop making new words for each place value after "thousand"? Rather than sticking with "myriad", a somewhat disused word for $10^4$, we call it "ten thousand", and then $10^5$ is "one hundred thousand", with no new word being introduced until "a thousand thousand", which we call a "million".

I suspect - and this is entirely speculative - that this happened because, in the time when this aspect of language was being developed, there wasn't much use for numbers as big as $10,000$, so they were described in terms of smaller numbers, rather than being named independently. Looking at the etymology of the word "million", it originally would have meant "a great thousand", which sounds a little less silly than "a thousand thousand". Note that, after that, the words for additional multiples of $1000$ use prefixes for $2$ (bi-llion), $3$ (tri-llion), etc.

I think a lot of it has to do with chunking.

For example: it's easier to remember (123) 456-7890 as somebody's phone number than 1234567890.