In equivariant homotopy theory, it seems like one tends to consider "genuine" $G$-spaces or $G$-spectra, rather than spaces (spectra) with a $G$-action.
My (rather soft) question is : why is that a reasonable thing to consider ?
For instance, a baby example of a difference between both is the (unique) $\mathbb Z$-equivariant map $\mathbb R\to *$. This is a weak equivalence in the model structure that produces "spaces with a $G$-action", but not for "genuine $G$-spaces", e.g. because $\mathbb R^\mathbb Z = \emptyset \not\simeq * = *^\mathbb Z$ (where $X^\mathbb Z$ denotes the set of fixed points for a $\mathbb Z$-space $X$).
What I'd like to understand is for instance why we might want that map not to be an equivalence.
Is it simply because we want an equivariant Whitehead theorem ? Or is there something deeper to understand here.
From my (very naïve) perspective, it seems like the naïve weak equivalences are very natural to think of, and preserve the most important aspects of $G$-spaces : the homotopy fixed points and the homotopy orbits.
My guess is that there are other types of "equivariant" structures that aren't well-behaved with respect to these equivalences, so it makes sense to require more - and I simply don't know enough equivariant homotopy theory (yet) to know about them. Is that the case ? What would be examples of such structures ?
Perhaps a related (broad) question, which might shed some light is : what are we trying to capture with genuine $G$-spaces (spectra) ?