From modern perspective this is much more straight forward than the "genuine" version you described above the question. Naive $G$-spaces are just functors $BG\to \cal{S}$ among infinity categories. $G$-spectra are just functors $BG\to \mathrm{Sp}$. You can think of a $G$-spectrum as a functor on $G$-spaces by $E \mapsto (X\mapsto \mathrm{Map}_{\mathrm{Sp}^{BG}}(\mathbb{S}[X],E))$ where $\mathbb{S}[-]= \Sigma^{\infty}$ is the stabilization functor, applyied pointwise to functors from $BG$. Hence, accepting some notions like functors and stabilization in infinity category theory you immediately get a theory of equivariant stuff of this "up to homotopy" flavour. In particular, if $E$ has trivial $G$-action then by trivial-colimit adjunction and colimits preservation of the stabilization we get $$\mathrm{Map}_{\mathrm{Sp}^{BG}}(\mathbb{S}[X],E)\simeq \mathrm{Map}_{\mathrm{Sp}}(\mathbb{S}[X]_{hG},E)\simeq \mathrm{Map}_{\mathrm{Sp}}(\mathbb{S}[X_{hG}],E)$$ and you indeed get the cohomology of the homotopy quotient.

In some sense, the surprising thing from this modern perspective is the existence of the "strict" version, which is slightly harder to define internally to modern homotopy theory, even thought it is duable.

From modern perspective this is much more straightforward than the "genuine" version you described above the question. Naive $G$-spaces are just functors $BG\to \cal{S}$ among infinity categories. $G$-spectra are just functors $BG\to \mathrm{Sp}$. You can think of a $G$-spectrum as a functor on $G$-spaces by $E \mapsto (X\mapsto \mathrm{Map}_{\mathrm{Sp}^{BG}}(\mathbb{S}[X],E))$ where $\mathbb{S}[-]= \Sigma^{\infty}$ is the stabilization functor, applied pointwise to functors from $BG$. Hence after accepting some notions like functors and stabilization in infinity category theory you immediately get a theory of equivariant stuff of this "up to homotopy" flavour. In particular, if $E$ has trivial $G$-action then by trivial-colimit adjunction and colimits preservation of the stabilization we get $$\mathrm{Map}_{\mathrm{Sp}^{BG}}(\mathbb{S}[X],E)\simeq \mathrm{Map}_{\mathrm{Sp}}(\mathbb{S}[X]_{hG},E)\simeq \mathrm{Map}_{\mathrm{Sp}}(\mathbb{S}[X_{hG}],E)$$ and you indeed get the cohomology of the homotopy quotient.

In some sense, the surprising thing from this modern perspective is the existence of the "strict" version, which is slightly harder to define internally to modern homotopy theory, even though it is doable.