How does one argue that the payment of a weregild is not punitive?
It's not
meant to be punitive, even if it has that as a side effect in some cases.
Punishment is about making the recipient
feel bad for what they did, while making the giver feel better. It's meant to control a person's behavior via shame and guilt.
Weregild is more about debts, and it only exists on the human level, within what's called the Inneryard. Taking something away from someone necessitates giving something back of equal value. A weregild is in place when one of the parties in this exchange is unwilling to make such a payment, or when the exchange is initiated without the consent of one of the parties (i.e., the "wronged.") This system doesn't care how people
feel about the whole situation.