"shoved his arms into the water up to his shoulders . . . When a fish swam between his hands, he smashed them shut"
Seriously, this needs some mythbusters-style testing.
The next thing I knew, he had grabbed a seagull by both feet with one hand and it was flapping it's wings wildly and squawking into my terrified face.
So I'd imagine that after a few weeks at sea and actually depending on catching birds/fish by hand for survival that one could do so.
I haven't caught fish with my hands but at sea been in situations where it didn't seem impossible. And the fishermen I know in the area (I live there on a boat) are magicians.
So sure, everything needs testing but I have a bit of a frame of reference around this one and it raises no red flags for me at all. Plus it's inline with the other lost-at-sea books I've read.
Someone else in this comments section linked to an article that explains how he caught the turtles. Basically (like you mentioned) turtles would get curious about the boat and swim right up to it, so he was able to get them that way. I presume the fish and birds behaved in the same manner, so it really lowers the red flags for me.
I would still love to see a video of someone catching these things the same way the drifter did though.
I guess my point is that even slippery little animals aren't as uncatchable as one would expect, if you can take them by surprise.
What's fascinating to me is that they start out as tiny little turtles that run like hell from the beach and swim as fast as they can. Most of them get eaten in the first couple days.