>then substituting one into the other, he derives (4.6)

(4.6) is what I called the simple oscillator.

As said I have no time for studying this, so I didn't look at the article. I hope someone else here on the forum might adress your questions concerning sloshing. Finally I am definitely not an expert for sloshing modelling or modelling in general.

Like intuitively for describing forced oscillations I would have rather started with an ansatz where the periodic forcing is in the constant term, like one learns in school i.e. something like $T_{tt}+\omega 2T= F(t)$, where $F(t)$ is periodic.