Why is the nucleus the centre of the atom?
Definition. Whatever the electron does, chemists define the position of the nucleus to be the centre of the atom meaning that electrons can be polarised away.
Quantum mechanics. When studying an isolated atom, the only thing that influences electron behaviour is the nucleus. You can calculate the ‘path’ of an atom‡ relative to the position of the nucleus with quantum mechanics and your result will be that quite a few electrons will occupy a perfect sphere whose centre is the nucleus.
The second point may require elaboration. But it really is quite simple. You have a very lightweight, very fast particle with a charge of $-1$ (your electron) and you have a very heavy, very slow particle with a charge of $+1$ (your proton). By the time the proton has moved an inch, the electron has moved a mile so we can approximate that only the electron really moves. (This has the fancy name Born-Oppenheimer approximation.) Now remember that both are attracted to each other. The proton doesn’t move, but the electron feels the ‘desire’ to ‘move towards’ the proton. (Inverted commas because this is a macroscopic view that only gets us so far when dealing with quantum effects.) If you average this over time, you will find that the electron will have spent the same amount of time on one side of the nucleus as on the other and the mean electron position will be in the nucleus. Spherically outwards from the proton, the chance of locating the atom decreases monotonously with distance; the angle does not matter. We have a perfect sphere with the proton being the centre.