Why is the blank symbol not considered part of the input alphabet of a Turing machine?
The main reason is that it allows the machine to detect the end of its input: it's (the character before) the first blank. If you allowed blanks in the input, the machine could never know whether it might find more input by scanning farther to the right. Of course, you could solve that by having a special "end of input" character but then you have to insist that that can't appear in the input, so you've just shifted the problem one level deeper.

It also makes the initial conditions much easier to specify: the input is the non-blank section of the initial tape, which must be finite and contiguous. And if you want a blank character to be a part of the input alphabet, you can always add an extra character (call it "space" or something) and have the machine behave however you want when it sees it.