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The best throw-L, bat-R player ever was probably Rickey Henderson.

Baseball Players Who Bat Lefty and Throw Righty Are More Likely to Be Great (mensjournal.com)

The study, which re-analyzed data from a similar 1982 study and incorporated new data from every major-league player from 1871 through 2016, finds that only 2% of the general population throws righty and bats lefty, but 12% of major-leaguers do so. In addition, 32% of the best-ever big-leaguers hit lefty but throw righty, which, the study authors argue, suggests that righties who adopt a lefty stance have a greater chance of becoming, well, great.

Cool, I'm the exception.  I was actually a plus defender, but the worst hitter on my HS team.  Turns out being left handed, but late on everything, just slapping it to left field with no power, is pretty useless

I think the only way to catch flies and judge them is to do it over and over.  I liken it to receiving serves in volleyball.   You have to keep seeing the ball come at you with different velocity, spin, arc etc.  The backdrop matters a fair amount as well.

Coaching my sons U9 travel team, it's tough.  We practice it, every kid can catch it if you throw it, no matter how high.  The way it comes off the bat is totally different, and none of us are good enough to consistently hit fly balls into the outfield.  It works like 1/4 of the time, then the other 3/4 of the time we miss, hit a grounder, or a line drive

Lol. Trust me, just like learning to catch a fly ball takes practice so does learning how to hit one.  When my daughter first got into softball I struggled to consistently get the ball in the air.  I can take her to the field now and hit 100 fly balls within a 10 foot  radius of each other.  Toss the ball up and hit it with an upper cut swing somewhere between your chest and shoulders.  You gotta catch it high.  If you let it drop below your chest you will hit a lot of line drives.

And it actually is important the kids are fielding fly balls off a bat because what you said is right.  There’s a world of difference in fielding a batted ball rather than someone tossing one in the air.  

Why is it different coming off the bat versus throwing it?  I agree it is, I don't understand why.  Spin? 

I think a thrown grounder is the same as off the bat, if the same speed.

I’m just spitballing here but I think there might be a couple of factors.  One is if I’m tossing balls in the air the kids have to be a lot closer to me.  I can stick kids 200 feet in the outfield and hit balls.  I can’t hand toss balls that far.  So, the closer they are the easier I think it is to watch the ball all the way out of my hand and pick it up easier.  The further away they are the more difficult it is to pick the ball up immediately.

I think there’s a moment when the ball comes off the bat that you lose it temporarily and then have to track it again.

I also think there is a world of difference in knowing a ball is coming to you as opposed to not knowing and having to react.  I like to put kids at all 9 positions and just randomly hit balls to them.  Not in any kind of order, just wherever, whenever.

Back in the day, I could throw a ball about as far as I could hit a fungo.  But it does look different.

I remember for fun at the end of practice some of my 12 year olds would stay late and I'd hit shots to them as hard as I could hit them, probably close to 300 feet.  Maybe 250.  It seemed like a mile. I was young and goofy.