Sophie, the reason for the calculus is only that it is not obvious that the field is zero everywhere inside a spherical shell. Knowing this means you don't need the calculus, but if you need to prove this, then I think you do need calculus.

I agree.

There is a fairly good 'arm waving' argument. Rather than calculating the force on you (sitting in the middle, somewhere), you think of the force on the surface of the shell (equal and opposite - so it's valid) and equate the gravitational, inverse square law, to the effect of ISL on illumination from a light source. Two cones with the same solid angle, pointing in opposite directions from you, will ' illuminate' the shell with an equal amount of energy. The light is spread out on the distant bits and less spread out in the nearer bits. The total light flux hitting each end of the 'double cone' will be the same, though, and, by analogy, the total gravitational forces on the two areas formed by the two cones of light will also be the equal and opposite. Forces on the shell and, therefore, also on you, will be balanced in any direction. It's easiest to visualise if you are on a diameter but it works for any position - you don't need to consider the actual shape which the light beams form on the inside of the shell.

(I made this up on my own and I'm quite pleased with it!)