Personally, I believe it for two reasons:
First, it seems that everyone but Musk is claiming so, so either everyone else is wrong, or Musk is wrong. I find it far more likely that this is just another "funding secured" type of claim, than an actual win. Case in point: see this evaluation [1] from only a year ago, where Tesla was ranked 19 out of 19 (Waymo was #1).
And the timing of the announcement, only two days before delivering really bad financial news... well, see the previous comment.
You provided the second reason yourself: Waymo has some incredible guys going after autonomy, and they've been doing that for a decade now. So whatever shortcut the Tesla people claim to have found, it's (a) either a shortcut nobody else has thought of before, or it's (b) they have thought of it before, and ruled it out. I believe (b) to be far more likely, hence I believe them to be on the wrong path.
Also, contrary to what you say, they do have something to show as of yet. They are operating hundreds of autonomous vehicles in Phoenix.
And yes, they are probably playing it super safe, but that's exactly what you should expect from someone accelerating 1+ tons of metal to lethal speeds, and giving it autonomy.
[1] https://www.teslarati.com/tesla-finishes-last-navigant-resea...