The big one is not retaining an attorney. No, he doesn't have to for small claims court, but Uber clearly had its legal team looking at this. Now, I understand that lawyers are expensive and carry their own issues - but if he'd won the case with an attorney, he could potentially have secured attorney's fees as well. Plus, a good way to make a company cave on an issue like this is to say "direct all requests to my attorney; I'm not speaking to you on this matter."
I realize getting a lawyer isn't always feasible, though, especially in small claims, so I'll assume it just wasn't possible for this case (also, I'm not a lawyer, this is just based on my understanding of the small claims process). Still, giving Uber all the evidence in the case after a suit had been filed but before any discovery had been ordered was a misstep. Small claims court rarely has formal discovery - the process by which each side finds and turns over evidence to the other side. The OP was - as far as I can tell - under no obligation to give evidence to the Uber rep of any kind. He should not have; they used it against him. What he should have done was say "we can discuss this in court, or you can make me whole now and avoid that," and NOTHING else.
Edit: A line I really enjoyed from the piece: "They were a company that shows blatant disrespect to authority, operating illegally in cities and using technology to intentionally avoid law enforcement." Yep. That's Uber in a nutshell. Well played.