First, let your customer know you are watching the situation carefully and they can be rest assured that if the package is lost in shipping, you will provide them with a refund.

I tell them that you can't open a missing package investigation for x days according to the USPS (if that's the carrier) but that as soon as you are able to, you will submit the form and get in touch with them.

Once you can submit the form, do it. It takes 5 minutes or less and you immediately get a case #. Let the customer know you filled out the form and the situation is under investigation, here is the case ID#.

If you want to score bonus points, you can even screenshot the email you get from the USPS with the case ID # and send that to your customer in Etsy's chat (you can upload photos).

For me, I sell incense. If someone buys a single little box of 12 incense cones and I ship it to them and it's been 2 weeks or more and the missing package investigation is going nowhere, I offer them a refund or replacement. I mean.. the cost to me including labor for that box is maybe $3-$4 + the cost of shipping is maybe $3.50-$4.50. It's a small price to pay to make my customer happy and to keep my reputation intact.

If you are shipping things that are $25, $50, $100 you should be doing this with USPS Priority or doing First Class and paying for the insurance.