I'm so glad to hear you got your phone working and were able to contact your cousin.
it's too risky to record your fingerprint where it could be used for any nefarious reason.
Your fingerprint itself is not being recorded. See here for an explanation of how TouchID works and why it is secure:
About Touch ID advanced security technology - Apple Support
That being said, there may be some fourth amendment issues around TouchID. The courts have ruled that an individual cannot be required to give up a passcode. But, the decisions are less clear on being required to stick out one's finger. So, definitely, your position has merit.
I would still personally recommend that you have a passcode on your phone but, in the end, that's up to you. If you're comfortable with the idea that any data on the phone could be viewed by anyone who picks it up, then, you don't need one. I keep a lot of stuff on my phone, including work documents that I would not want other people to have access to (and no one really needs to know how many pictures I have of my cats).
Best of luck.