I suspect they wanted to sue for a large payout, much more than covering health care costs. Notice their attorney refused to discuss the proposed amount. They figured a jury would give them more than an arbitrator paid by Uber.

This is pure speculation and unhelpful to the discussion. This sets a bad precedent for the reach of admissibility of T&C agreements, regardless of amount.

Just uber? Doesn't that practically apply to any app you use with any corporation?

Yes, of course you are correct. There was funny South Park where Apple confiscated the kids because the terms of service had signed away their life.

But in this specific case. Guess I'm now faced with this actual bodily harm, and it isn't a joke anymore.