I've never damaged/lost any phone so far, from my Droid X, to my S4, S7, S10... 🤞I even have a working Microsoft Lumia 640LTE, lol.
but this Z Flip looks like it needs some warranty protection.
I am sorry to hear your Flip 3 is having difficulties enrolling in Care+, and can understand the concern, as I am clumsy with my devices and like phone insurance on mine. You would need to speak with our Samsung Care+ team, who would be able to assist you through this and get more information regarding this particular case and to get a resolution. The number to speak to them directly is 1-866-371-9501.
Please let me know if this helps, and I hope you have a great evening.