There are differing schools in terms of skepticism. There is the absolute skeptic, one who believes that no principle can be proven objectively, and the moderate skeptic, a person who simply argues that matters are to be questioned before being added to objective perceptions. The second of these is the more traditional and well known skeptic of which you have already mentioned. He or she is pragmatic and operates in a manner completely sensible with the world around them. The first however is an abstract entity that doesn’t even exist. He is a nihilist in the purest sense of the word and by even admitting to himself that he is indeed a skeptic, he violates the rules of absolute skepticism by admitting to a possible fact, “I am a skeptic”. But due to the fact that to even be a skeptic, one must admit to the objective fact that there are no objective facts, to be a true skeptic vacant of any assumptions is impossible; man is bound by a definitive mindset simply by existing in this world or more specifically simply by processing it. That is not to say that this is a bad thing, however. Without an acceptance of at least some definite facts it would be impossible to form an identity or even a sense of self to relate to others, the main form of thought that human beings uses to understand others in the first place (“What if I were in their shoes?”). So while their “ability to percieve the world through the eyes of their fellow humans” may be sharply diminished this would also be true if we were free of definite assumptions (“I am a person”, “I am an individual”, “There is actually something to understand about the self and others”). A complete erasing of certainty is neither desirable nor possible though its success would surely lead to Nirvana. A complete deletion of the ego, to accept not even the basic premises about the self or the world around the self as definite or provable, to exist in nothingness.
By the way, you’re correct in that logic can be incredibly restricting. Barring some very basic assumptions that we need to make to use it, the actual manner in which we go about logic is completely dependent on our perceptions. For example, in order to find that 1+1=2 in the first place, we must first understand that the variables we are working with are 1 and 1 as opposed to say, 1 and 2. This is a product of our perceptions and as our perceptions can be wrong, the product of this logic can be vastly disarrayed. A nihilist would just say that the workout of every logical problem is x+y=z since the variables they are working with cannot be definite.