I agree. Be very careful working him on barrels with the ground you have -- even just trotting. That ground is VERY hard with no footing whatsoever. He can very easily slip; even at the trot.
He does look like he opens up and streches quite nicely when asked to run. Looks very similar to my gelding, who is also a paint (but no color) and is also 16.1 hands. His turning style is different than most and he looks slow in his times because he is so big, but he sure clocks time! We won the 1D a few times this summer which is the best we've ever done. I was so proud of him! He just eats up the ground when he runs because he's got a huge stride.
So I think he looks like he would do very nicely for barrels. And it sounds like you've got a nice slow, solid start on him if you are still only trotting after 3 months and that's SUPER! He's going to have a solid pattern under him and not be soured or ruined by being pushed too fast too soon.
As far as poles, if he's like my horse, his size is going to hurt his poles performance. It's a lot easier for a shorter horse to have time to collect up and run through those poles (at least in my experience). However, every horse is a unique individual so you never know until you try!
This is an old picture of my horse and I running, but I like it because you can see how much he is completely using that 16.1 body of his and stretching out!