Barzal has had hundreds upon hundreds upon hundreds of opportunities to demonstrate that he will do all of those things — take the puck to the net, attack the net, take his shot if it's there, yada yada — and he has simply not done it. He has many gifts and is insanely talented, but he repeatedly turns away from A+ goal opportunities and will only shoot the puck as a last resort, often to the detriment of the team's offense.
The idea that he is forbidden to try to score goals by Trotz and now Lambert, that one day the coach will say "OK, you're unshackled now" and he will suddenly pump out points like Jack Hughes or MacDavid, is one of the dumbest claims that consistently floats around this board. Why would the coaching staff insist on having the most dynamic offensive player not try to score? But others can? Zack Parise outscored Barzal last year (goals, not points). So did Palmieri, in fewer games. Hudson Fasching, for God's sake, was right on his heels in fewer games.