I can say with a lot of confidence that I watch Barzal more than you ever could. Your analysis of his style of play is not correct. He does not stand in front of the net because he is very much a possession player, but he does drive to the net with the puck and does not shy away from contact. He is most definitely not a perimeter player. At the same time, Nylander will also drive to the net (see his goal in game 5 against Florida), but Toronto fans love to paint Nylander as a soft player because someone has to be the whipping boy.
Switching Barzal and Nylander would be an interesting experiment, because I suspect you would see their point totals flip, as well. Barzal would score far more points in Toronto's lineup because of lineup and power play quality, and Nylander would score fewer on the Islanders for the opposite reason. It would be a meaningless trade for both teams because exchanging these players wouldn't fill a need for either team.
I doubt very much that Toronto will trade Nylander. I expect that both he and Matthews will be signed on the eve of training camp, which will be the next pressure point in these negotiations.