Not only nucleus charge grows, also number of electrons shielding the charge.
But there is something much more important. Electrons don't form an amorphous cloud around the nucleus, they are ordered in a way and occupy orbitals. That basically means that new electrons go further from the nucleus (oversimplification and not exactly true), so the attraction is not higher for each next element.
Once you will learn a little bit about quantum chemistry, quantum numbers, orbital energies and orbital arrangement in space it will become more clear.
Although Borek refers to an oversimplification (and not exactly true), it is probably a lot more useful than any other treatment I can think of. I don't want to go over my head in a discussion here, but even quantum mechanical calculations must remain consistent with the principles of physics. (I think that could have been construed as a point Einstein was making in the Bohr-Einstein debates, but that would be another day.) Even with a simplistic model of atomic structure, it will come reasonably close to the physicochemical properties.
"Electrons don't form an amorphous cloud around the nucleus, they are ordered in a way and occupy orbitals." Agreed! Ice is tetrahedral, therefore the electrons or electron pairs are concentrated in defined regions. If you compare the bond lengths of LiH, BeH2, BH3, CH4, NH3, H2O, and HF, the bonds become shorter and consistent with an increase in the nuclear field. In reactions, LiH, BeH2(?), and BH3 are hydride donors and NH3, H2O, and HF are proton donors. I interpret this as an indication of the relative shielding effect of the inner electrons. If you compare dilithium (267 pm) and difluorine (142 pm), their distances appear consistent with the inner electron shielding, a nuclear charge effect, and the inverse square law.
Even if a simplistic model does not result in a correct quantum mechanical calculation, it does seem much easier to grasp and would correctly predict the nuclear field of sodium would be much smaller than elements to the right of it in the periodic table. The inverse square law also predicts the field would be much smaller and thus reinforcing its low affinity for additional electrons.
I think the difficulty in understanding the effects centers on how to treat cations. At large distances, we can treat them as Gaussian shells and only consider their net charge. At short distances, we must be cognizant of their microscopic properties, specifically, the electrons are negative and repel other electrons. A net positive charge does not convert electrons into positive charges.
I do not wish to highjack the posters question, but I urge caution in the accepting some tenets of quantum theory. Even orbitals have unanswered questions. Pauling introduced hybridization to achieve tetrahedral structures while s, p, d, and f orbitals describe the atomic emissions. I am not aware of sp3
emissions nor how strict adherence to s and p orbitals should give a symmetrical methane. Even Gilbert Lewis was skeptical of the Bohr interpretation of the emissions of hydrogen.