It looks as though my question might turn out to be a duplicate. If so, it does not need an answer, after all, thanks.
ORIGINAL QUESTION
Why does convention recognize $\pi\approx 3.14$, rather than $2\pi\approx 6.28$, as the fundamental quantity?
You have some contour integrals that come out to $\sqrt{\pi}$ or $1/\sqrt{\pi}$, so that's kind of neat; but I don't know that that arises more often than, say, $1/\sqrt{2\pi}$, as in the Fourier transform. Anyway, all that special-function action is so advanced that it misses what might seem to some to be the main point: $2\pi$ is a circle. How much more fundamental can you get than that?
But mathematicians are smart people (and I am just a building-construction engineer), so I do not doubt that a good reason exists. I just do not know what the reason is. Hence the question.
Why wasn't some other symbol defined, $\kappa\approx 6.28$?
The short answer to your question is that there is another symbol for $2\pi$, namely $\tau$. Some people promote that notation, but it hasn't caught on in the mainstream of mathematics, at least yet (and I don't think it will).
The longer answer to your question comes from history. The reason that $3.1415...$ got its own symbol $\pi$ before $6.28...$ did is that $\pi=3.14...$ is the ratio of the circumference of a circle to the diameter of the circle. Finding that ratio was a question that Babylonian, Chinese and Greek mathematicians/geometers in antiquity would quickly encounter while studying geometric shapes. Measuring/approximating that constant became an interesting problem for them. Mathematics had to develop much further before the value $2\pi$ would begin to show up often enough to conceivably merit its own symbol. The value $2\pi$ started showing up frequently only after the idea of the radian measure was invented in the 18th century, because then it became natural to think of a full $360^\circ$ rotation as a rotation of $2\pi$ radians. Then as complex analysis, fourier analysis, etc developed, it would show up more often.
I have no definite answer as to why people have not started using $\tau$ or some other symbol instead of $2\pi$. I suspect that simply there is no real need for it; $2\pi$ is easy enough to write as it is.