New return kit with iPhone 16... looks flimsy... reusing the box it came in?

The boxes have always been a bit worrisome in my opinion but have personally never had a phone damaged in transit using it for trade in. If you have an Apple Store that you can go to you can opt to complete your trade in there versus mailing it in. Usually the label for the trade-in is already affixed to the box and has info that will make it easier for them to properly process your device once received. Opting to use a different box without that label will pose a risk to things going smoothly but you can always contact Apple directly first to see if you have other options

I would advise against sending it in and take it to the Apple Store directly if you can. Of course this depends on how close the nearest store is to you. Many times the value of the condition is subjective and the owner values it much higher than when it is evaluated after being sent it. It is possible that they will come back with a lower trade-in value and that becomes a huge problem. By taking it to the store, you will get an immediate value for the phone and no surprises. You also won't have to worry about any damage went sent back or having it get lost/stolen by the carrier. Depending on how far away your nearest Apple Store is, the extra drive time may alleviate a lot of problems.