For something like an irreplaceable heirloom I'd prefer to deliver it in person.
However, for the sake of discussion, if I had sent something and it hadn't arrived after five weeks ... or even five days, depending on what kind of shipping I had chosen ... I'd be raising major questions with the carrier.
I have never shipped anything commercially, only received things as a shopper, and while problems are rare things do "go missing" sometimes.
In my location, the post office is by far the most secure way to receive things.