I would be more concerned about them receiving your phone then making up an excuse to cut the trade-in amount in half, or even to zero. That is what happened to me with an iMac trade-in.
dropped my 13PM trade-in for 17PM off with Fedex on Wednesday, got an email an hour or so ago from Apple saying it's all good and got my $320.
I've done several iPhones and watches this way and never had an issue, but YMMV
Although my personal preference is to process my trade in the store, I have shipped many of the families trades using the kit, or packaging that Apple mails out.
The key is to patiently study the instructions and carefully fold the package. You end up with great protection in a very secure enclosure.