Given that Sp is better behaved than all other existing models of spectra
No, Sp is not better behaved than other models. The reason that it seems to be is because all operations in Sp (e.g., Ω^∞, Σ^∞) are (by construction) derived. But if you derive the operations in any other model, you will get exactly the same properties.
are them still needed for the purposes of homotopy theory?
As stated, the answer is obviously no. (The underlying (∞,1)-categories are equivalent, and arguably, this is the only thing that homotopy theory cares about.)
But if “needed” is interpreted in the practical sense, then one can point out many situations where working with strict models is much easier than with quasicategorical models, and because of the various rigidification results has exactly the same generality as quasicategorical analogues.
This especially concerns all sorts of situations that in the quasicategorical world involve (co)Cartesian fibrations, which includes, in particular, the treatment of monoidal categories and operads.
Quasicategories appeared (in HTT) when other models were much less developed (e.g., the foundational paper by Barwick and Kan on relative categories has not appeared yet), and today they are just one model out of many, which has exactly the same foundational status as relative categories, simplicial categories, etc. Some models, like complete Segal spaces, have better theoretical properties, other models, like relative categories, have a much better supply of examples.
In fact, quasicategories are typically constructed out of simplicial categories or dg categories (typically with a model structure, see Lurie's books for many examples), and given the fact that the underlying (∞,1)-categories of all these models are equivalent, one may wonder why one should bother with passing to quasicategories at all.
To summarize, quasicategories are themselves a model: a quasicategory presents an (∞,1)-category as an (∞,1)-colimit over Δ^op of (∞,1)-categories given by finite chains of composable morphisms. Such a presentation is often convenient to work with, in particular, when doing theoretical computations. But it is merely a presentation, and as such it will inevitably become inconvenient at least in some situations (such as those that require extensive use of (co)Cartesian fibrations). Why cripple oneself with just one model, when other models can be much better?