Gauss’s Law applied to an infinite sheet of charge

1. The problem statement, all variables and given/known data
The problem that inspired the upcoming question is: Find the magnitude of the electric field due to an infinite sheet of charge with uniform charge density σ using gauss’s law.

I have in fact arrived at the character answer σ/2ε (epsilon nought), but I don’t understand this solution. To elaborate, I’m unsure as to why I was able to solve this problem by using a cylinder of area A that only encloses a small portion of the sheet of charge (similar to every example online of this classic problem). Why does this small Area A represent the E field for the entire sheet? I’m also wondering why I cannot use a sphere and symmetry to solve this problem; Is the E field not constant?

2. Relevant equations

electric flux = e * da;
flux = charge enclosed/epsilon nought
area of ends of a cylinder 2*pi*r^2

3. The attempt at a solution

I have found the answer using a cylinder, but I don’t see why this works, or why I can ignore part of the charge of the sheet; Or in other words, why charge enclosed in the above equation need not be equal to the charge of the sheet, but that of a tiny cross sectional area.

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