This question asked whether $\mathrm{Sp}$ is convenient in the sense of satisfying (in the $\infty$-categorical sense) a list of desired properties of Lewis in his 1991 paper (see there).

The answer given there is yes, provided one interprets Lewis's desiderata in the setting of $\infty$-categories.

Given that $\mathrm{Sp}$ is better behaved than all other existing models of spectra, are them still needed for the purposes of homotopy theory?

This question asked whether $\mathrm{Sp}$ is convenient in the sense of satisfying (in the $\infty$-categorical sense) a list of desired properties of Lewis in his 1991 paper (see there).

The answer given there is yes, provided one interprets Lewis's desiderata in the setting of $\infty$-categories.

Given that $\mathrm{Sp}$ is better behaved than all other existing models of spectra, are them still needed for the purposes of homotopy theory?

Here $\mathrm{Sp}$ means the $(\infty,1)$-category of spectra, not specifically the quasicategory of spectra.

(@Dmitri Pavlov requested clarification on the meaning of $\infty$-category. I meant quasicategory, but only because of ignorance: I don't know constructions of $\mathrm{Sp}$ other than Lurie's, and would be happy if someone could point me to references on this. Regarding the meaning of “$\infty$-category”, I believe the discussion would be more interesting if we do not restrict to quasicategories. (This should be a comment; see the edit summary))