>But say, if he hadn't been sent to prison, would he have paid child support then?

The amount could have been ordered to be taken from his bank account, or confiscated from his salary directly from his employer or payment system, or taken from his property.

There's absolutely nothing that necessitates that he is sent to prison if he merely can't pay.

Only if he has money and doesn't pay (and that after accounting for his food and rent expenses at a nominal minimal fee), or hides money that he could have used to pay.