irrational numbers - Why are $e$ and $pi$ so common as results of seemingly unrelated fields?

I'm sure this gets asked all the time but I swear I googled with no useful result. What I'm looking for is a reasonably intuitive answer.

Those two constants have some pretty interesting properties. $\pi$ is often used in geometry while $e$ is for example used in statistics, yet $\int_{-\infty}^{+\infty}e^{-x^2}=\sqrt \pi$

Weird right? We'll it's weird to me.

$e$ itself is pretty unusual. It's both the $\sum_{n=0}^{\infty}\frac{1}{n!}$ and the limit of $(1+\frac{1}{n})^n$. Personally, I've done proofs that those two converge to the same number but I really can't say that I get why.

Then there's the famous $e^{i\pi}=-1$

This can't all be a coincidence, right? In fact, seeing how there's not only an infinity of irrational numbers but an uncountable one, this literally can't be a coincidence.

There must be some level of dissection on which you can say "See, there's e. If we add 1 to this we get $\pi$".

For example, say we turn the tables. $\cos(\pi)=-1$ and $\frac{1}{e}\sum_{n=0}^{\infty}\frac{1}{n!}=1$. I could say it's pretty weird how these unrelated things have those numbers as their results. $-1$ and $1$ pop up everywhere, right? The intuitive explanation is that I obviously chose those two things specifically because I know what their results are. As for what their connection is, they obviously come from the same "factory". They are both whole numbers that you get by slightly incrementing or decrementing zero.

In fact, I'd be willing to bet that whole numbers with their absolute values less than, say, 10 pop up pretty often and the explanation for that is the same as above.

So what's the deal with those irrationals? They aren't countable. You can't just add one to one of them to get the other. What's their "common factory"?

The gist of what I'm saying is, $e$ and $\pi$ show up all the time in unrelated circumstances and they even seem to know each other. Their connection, however, eludes me.