How can the size of atom be the cause of octet expansion?
Your revised version of Atkins’ statement is correct, no hypervalent main group compounds require the involvement of d-orbitals. Rather, all of those molecules formerly explained by d-orbital participation can just as well be explained with 4-electron-3-centre bonds (including resonance structures) and charge separation (e.g. in sulphate; no need for resonance), giving rise to all-octet Lewis structures. In fact, calculations have shown the d-orbital participation to be minimal and neglegible at best.

A much more relevant question is whether more than $x$ substituents actually fit around the central atom. This is especially notable for the difference between the second and third periods: The reason why nitric acid is $\ce{HNO3}$ while phosphoric acid is $\ce{H3PO4}$ is likely due to the much smaller size of nitrogen, not allowing for tetracoordination by oxygens in the $\mathrm{+V}$ oxidation state while the larger phosphorus atom can support four oxygen atoms. If I remember correctly, that was also discussed as a reason why $\ce{NF5}$ hadn’t been found at the time of Klapötke’s article while $\ce{PF5}$ has been known for quite a long time.

It gets more pronounced when you go even lower in the periodic table, i.e. to higher periods. Antimony is known to be coordinated by six oxygens in the $\mathrm{+V}$ oxidation state ($\ce{H7SbO6}$, if you wish, although $\ce{H[Sb(OH)6]}$ is a more accurate description) — something phosphorus could technically do ($\ce{PF6-}$ with small fluorine atoms is known) but usually doesn’t. It is mainly the greater atom size of antimony that allows the coordination of two additional oxygen atoms.