Why is it correct to use the sqrt to cancel out the sqrt operations in standard deviation?

Well, (a+b)^2 does NOT equal a^2 + b^2, rather it equals a^2 + 2ab + b^2.
Closely related, √(a+b) does NOT equal √a + √b, i.e. √ does NOT distribute over addition. Similarly, √(a^2 + b^2) does NOT equal a + b.

Example: √(3^2 + 4^2) means √(9+16) and since 9+16=25, you get
√(9+16) = √25 = 5. So, you would be wrong if you said √(3^2 + 4^2) = 3+4 because 3+4 = 7, and 7 does NOT equal 5.

This is just basic algebra and really has nothing to do with standard deviation.