I've been having the same issue on my kid's Amazon Fire HD 8 tablet. I fixed it by opening the Settings app and manually installing updates. After updating the OS and restarting the tablet, I could change profiles and access the pull-down notification/settings bar again.
Spoke to Tech Support, they tried everything but they ended up saying a factory reset was needed. I wasn't prepared to do that as although the device is 2 weeks old my son would not want to loose his games (again!) etc.
So, I tried a few other options. And, I found a fix that worked (for me at least):
I cleared the cache and stopped the AppStore and "Fire for Kids" (I think this is called something else in the US).
Turned off Alexa
Rerun "Fire Device Setup". I didn't change anything, just opened and clicked continue.
And, I have access to the profiles again. I hope this works for others and time will tell if this remains fixed. Clearly a bug that needs fixing and I had my issue flagged to the Dev team.
Totally worked even without the “fire device setup” step. Thanks!