I admit, that video makes me wanna reconsider
That video made it very clear to me that my position on the issue would be unwavering. My young DD seeing the video confirmed that were are on the same page.
Arguments I've heard (from coaches, mostly)...
"The mask makes it harder to see passed balls."
A hockey goalie spends a significant amount of time looking behind him at action behind the net, and moving quickly from one side of the ice to the other. They don't seem to have an issue tracking plays. A hockey-style catcher's mask does not inhibit peripheral vision or head movement. Also, once you've scooped up that passed ball, and you go to make the tag at the plate...well, watch the video again...
"The mask makes is harder to see throws from the fielders".
Why is it harder to see a throw from a fielder than a pitch? It's not. Leave the mask on.
"Pros take their masks off."
Not all of them.
(There are LOTS of examples)