It's obvious that some part of the 5s caused low hardware yields, probably either the A7 chip or the fingerprint scanner. Also keep in mind that this is the first time they've sold in China on Day One as well. I don't believe this was a case of artificial constraint of supply; I think Tim just couldn't get enough units made in time.
You should write Tim if you really feel upset about this, but I'd suggest you write him only when it's something actually important*, not just because you have to wait a little while for a product. It's just a phone man, and your daughter is taking it well.
(*) - I wrote Tim and suggested to him that Ive become head of design, this was three months before Forstall was ousted. I'd like to think my letter had some effect. I know that Jobs (or his assistants) read my letter during the original iPhone launch when I couldn't get mine to activate for three days, as a bigwig at ATT contacted me and made everything right. So they do listen, but it's not something to abuse.