I think having Fury a broken down old man was - far and away - the best narrative choice. You want your protagonist to start a story on the back foot, as it adds tension to the outcome. Something like John Wick is only really palatable in small doses, and would be boring with overexposure.
The main issue with the story is they had a great setup for a paranoid spy thriller, what with shapechanging aliens and Nick Fury, and completely squandered it. I realize it was likely for budgetary reasons, but it was weird as hell that the Skrull kept to one, and only one, face 95% of the time, and the one imposter who meant anything on the show (Rhodey) was revealed so early. They could have played around so much with Fury not knowing who to trust due to anyone having the potential to be an imposter!