How do parenthetical if-statements work? E.g. "he is a happy, if quiet, boy."

Try rearranging the sentence like this:

"Although he is quiet, he is a happy boy."

The word "although" shows that you need a contrast there. This if statement is simply another way of ordering the idea. It's a question of style. That sounds like something you would find in a book, a narrative style. But it could be speech. Imagine:

"Tyler's entrance to the race is a welcome, if unexpected development..."

Notice we're still talking about a third party, like commentary (narrative style). If I reword it:

"Tyler's entrance to the race is an unexpected, but welcome development..."

It now sounds more like speech and less stilted.