Is it at all idiomatic to say "between doing something and doing something else, I've realized that..."?

You can use the idiomatic expression what with:

What with learning and teaching, I have realized ..

(Cambridge Dictionary)

This is a correct locution for expressing what you have in mind, as shows the folowing OALD definition.

​between doing something - used to show that several activities are involved
Between working full-time and taking care of the kids, he didn't have much time for hobbies.

There is, however, another one, which is "what with" (it's the object of user 121863's answwer). I couldn't assert which of the two locutions is more common or preferred but it seems to me that "what with" would be somewhat more expressive, maybe because of its carrying no ambiguity.

I am going to disagree with the highly upvoted answer by @LPH

Between refers to a place that is neither one thing nor another.

Between working full-time and taking care of the kids, he didn't have much time for hobbies.

This means that there was no time or space between working and caretaking to have a hobby.

between learning and teaching, I've realized that...

This could mean that you took a break between learning and teaching and it was during that gap that you realized something.

At best it is ambiguous.

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Suggestions

Through learning and teaching, I've realized that...

Having experienced both learning and teaching, I've realized that...

Having been both learner and teacher, I've realized that...