Im not gonna say he isnt affected since you the player arent for gameplay reasons, Id have to say in universe Big Boss misses that eye and is at least somewhat worse off for it. I brought up finishing MGSV because (spoilers for the end follow)
Big Boss never even loses a limb its venom snake that does
The strengths and weaknesses of a character should naturally affect how the player interacts with the game. Gameplay and story are not completely separated for no good reason. Old Snake doesn't have a disability, but his old age and the wear and tear on his body actually did have an effect on how you played the game. Rolling for example might not execute properly due to snake's bad back. It wasn't major, but it was enough to give you the good impression that you were playing an old character who was on the verge of death.
Big Boss losing his eye and arm feel like just cosmetic choices. BB can aim just fine without is eye and he does the same soldier stuff as before with the bionic arm. The new arm gives him a ton of advantages as well. I don't think the games ever bring up Big Boss and his missing eye after the fact, so we don't know how he feels about it. He seems fine without it honestly. MGSV at least brings up the feeling of loss when BB loses his arm. It feels a bit hollow though considering he isn't put at a disadvantage in how he operates on the field.
Venom Snake/Big Boss share the same title. Yes, they're technically different characters, but when I say that Big Boss losses a limb, people know that I'm talking about the BB from MGSV. I find it preferable just to say Big Boss so I don't have to spoiler tag stuff.
In the original Drakengard all of the characters lost something in exchange for making a pact with some kind of creature. The main character loses his voice and becomes mute, another guy becomes blind and the final two party members lose their fertility and ability to age (as a six year old boy), respectively. The latter two are a bit debatable, but the former two are definitely disabilities.
"As a child, a young boy named Gaou lost an arm and an eye, and as a result, from a young age he was discriminated against and teased, unable to fit in with his peers. The constant ridicule and persecution caused Gaou to become bitter, and hardened, eventually turning to a life as a thief."
Duster from Mother 3 has some sort of unspecified problem with his left leg which causes him to constantly limp when moving. However aside from that, he's able to move fairly normally, still able to run and kick.
Was gonna post this, despite his bad leg he's a badass, specially when you realize his basic attack is kicking and he has a special skill where he back flips into a special kick to prevent enemies from gaining initiative by striking from behind you.
Ignis is a pretty good example.
I know a lot of people were annoyed by the level where you have to guide him and he is holding the whole group back and slowing down the pacing.
But I liked how it incorporates the fact that he is blind now into the gameplay. Unlike 99.9% of other "blind" videogame characters that are only realky blind in the lore but they act and play as any other character.
I know some people here dislike FFXV but the animations created for Ignis because of his disability was some of the best things I've seen in videogames. It is one of the reasons I liked FFXV despite some of it's problems.