The question isn't simply a "yes or no" for believing in fully autonomous driving -- at least not for me.
It comes down to timeline.
Do I believe Elon has a good chance to achieve it ? (without a specific deadline attached to it) Yes, I do !
He has thrown the best engineering talent at it and spares no expense to pursue it, so if any team can do it, his team will; and I do believe the problem is solvable.
On the other hand, I am skeptical about the "very soon" part. Sure, they are making great progress. I even believe that they will release some version of FSD to the wide public this year. But that version will still be a level 2 ADAS, not fully autonomous driving.
How many more iterations (software and hardware) will it take to make it true "FSD" (remember what the acronym stands for!) I have no clue, but my gut feeling is that it will take many more than what Elon currently projects. There will be many more gotchas just like there was many in the past 6 years of Autopilot development, don't forget his projection about "driving from coast to coast on autopilot" within a year from 2016.