While the charm of Captain America is that he pushes the boundaries of human potential without being truly 'superhuman,' Marvel's decision to give Sam Wilson a very specific version of Wolverine's healing factor was definitely the right call.
In 2012's Captain America #25, Sam Wilson became Captain America, taking over the role after Steve Rogers was depowered. Along with his falcon Redwing, Sam fought villains including Scarecrow and Baron Zemo, briefly leaving the Captain America title behind before returning to it recently (coinciding with his adoption of the role over in the MCU.) One big difference between Steve Rogers' Cap and Sam Wilson's is that Sam boasts a huge pair of mechanical wings. Unfortunately, unlike Captain America's fabled shield, these wings can be broken.
That happens in Avengers: Rage of Ultron by Rick Remender, Jerome Opena, Pepe Larraz, and Mark Morales. When Ultron returns to Earth with a vengeance, the Avengers stand against him, only to be faced with robotic duplicates of their classic roster. When Sam takes on an Ultron-ified Scarlet Witch, her powers shatter his wing, leaving him dependent on Spider-Man to escape. Sam manages to take possession of a weapon that can kill Ultron and is picked up by the Avengers Quinjet, only to then fly to safety with his own wings when the vehicle explodes. When Spider-Man questions how Captain America's wings are once again functioning, Sam confirms that they're made of self-repairing vibranium and have already healed from the damage caused by Ultron.
While giving Sam Wilson's wings a Wolverine-like healing factor is a drastic move for Captain America - who is often depicted as a low-powered character who nonetheless keeps up with gods and aliens - it's the right decision for the character. Sam's wings being as unbreakable as his shield would make him overpowered as Cap, while having him need to repair his wings whenever they're clipped by a villain would take away from the fun of his character design and signature flight. In contrast, having wings that can be damaged but then heal over time is the perfect compromise. As Rage of Ultron shows, Sam can be shot down by an enemy, struggle to escape them on foot, but then regain his flight capabilities once it's convenient to the story.
Wolverine's healing is an iconic aspect of the mutant brawler, allowing him to be hurt - often badly - and then return to the fray once it serves the story for him to do so. Applying the same logic to Sam Wilson's wings (though crucially not his body, like Logan) is the neatest way for Captain America's flight capabilities to be whatever the story calls for in the moment. Of course, Sam Wilson's wings being able to heal like Wolverine also addresses a longtime 'plot hole' often pointed out by fans - in a world where Captain America is friends with geniuses like Black Panther, Iron Man, and Reed Richards, it makes sense for him to be wielding technology a little more advanced than an unbreakable discus.