the PAY amount is set on the PAYMENT. if you don't change the payment amount, the amount to pay won't change.
that is normal behavior...

Logically, when I add items to an invoice this should automatically update the payment. It updates the invoice total in items, so why then does it not update on the payments tab as well? It looks silly when the client gets the invoice and it shows all the items and the correct total until you get to the payments pending and it shows the incorrect initial amount.

No, not logically.
The payments are meant so the admin can set them however he wants, and he can create multiple.
This is how it is supposed to work.
The fact that this would make sense for your website does not mean it would work for other users. What if an invoice contains multiple payments, and some are paid but others aren't?

Again, this is how the extension is supposed to work. Payment are NOT modified on a modification of items in the invoice. if you want to update a payment total, you have to input the amount.

Thanks for the explanation. So out of curiosity then, say I have an invoice that was created like I did and added additional items after the fact. If the client then pays the invoice total that was set prior to the modification, would the remainder then become payable or how does that work from that point?

no. again, you have to create the payment for the remaining amount (or change the value of the original payment to reflect the new amount)