Why is π irrational?
That's a strange question! Why would it not be? In a very precise mathematical sense, "almost all real numbers are irrational."
You can find Ivan Niven's proof at https://projecteuclid.org/euclid.bams/1183510788
It took over 2000 years of work by the best mathematicians in the world to show that there are any irrational numbers.
It took the best mathematicians in the world another 2000 years to demonstrate that $\pi$ is irrational.
Despite Halls's citation, I fear there is no easy way to answer your question. It would not have taken 4000 years to show that the ratio between the diameter of a circle and its circumference is not a ratio between whole numbers had it been an easy question.
Why "despite" my citation? That proof is only a page long. And when do your "2000 years of work by the best mathematicians in the world" begin? The proof that $\displaystyle \sqrt{2}$ was irrational goes back to Euclid.
(Were you thinking "transcendental" rather than irrational?)
Your cited paper is indeed brief, but brevity does not entail simplicity. The proof that $\sqrt{2}$ is irrational is simple and, as Hardy pointed out, incredibly elegant. Your citation is far more sophisticated in applying integral calculus to trigonometric functions. It obviously assumes a fair amount of mathematical training. The OP was entered in the New Users forum. The assumption that your citation is comprehensible to someone who has never posted here before strikes me as unwarranted. I doubt you would circulate it to a high scool algebra class in the expectation that it would do anything more than silence inquiry. "Shut up kid and listen to your betters."
As for the history, Greek mathematics followed upon millennia of mathematics in Mesopotamia. It is true that virtually none of the results of that work have survived, but there is definite physical evidence that the Greeks were not the first mathematicians. Based on perhaps faulty memory, the Rhind Papyrus precedes Euclid by at least a thousand years and was itself preceded by cuneiform tablets from Mesopotamia. The proof that $\sqrt{2}$ was irrational was highly disruptive to the Greeks: it was not something that they were aware of from elsewhere in the Near East.
As for $\pi$, Archimedes bounded it with rational numbers, and I suspect that from then on $\pi$ was conjectured to be irrational. My point was that proving that conjecture took mathematicians a great deal of time, many, many centuries. Calculus was developed in the late 17th century, but your proof is dated almost 300 years later so I guess it was not obvious to Newton or Euler.
The simplest proof that pi is an irrational number from what I've seen is that the number pi is equal to half the cosine of zero, and then by the method of proof from the opposite it follows that pi is not the result of division of integers.