I checked given json and found no links to videos, just UIDs. So, yeah you helped me to get unauthorized access to videos i was not intended to get. Now, it all depends on your jurisdiction if its punishable or not.
You're pretty clearly being a bad actor here.
Links to videos on their site that you need to log in to view? Totally no problem and they'd be unreasonable to complain.
Deriving direct links to private content to make it accessible to the general public? Doesn't matter if their content has the security of a wet cardboard box, you should not be doing this.
I think they should probably be fixing deep linking to the content working for unregistered users instead of threatening you with legal action, but that doesn't make what you're doing OK.
I don't think you have a guilt-free position, the fact your website is public puts you at risk of litigation (not a lawyer.)