(This is an answer to a question of below from Akhil Mathewthat I can't seem to find; he wanted examples of ``explicit computations'' since all he know knew were classical 1950s calculations and abstract theory.theory. My answer is too long for a comment and too short to do justice to the question.)
That is terrible!!! I don't know where to begin, since there are huge masses of explicit calculations in the past half century. You mention infinite loop spaces. There is a theorem that says that the ordinary mod $p$ homology of $\Omega^n\Sigma^n X$ is an explicit functor of the ordinary mod $p$ homology of $X$, with all information (product, Dyer-Lashof operations, coproduct, Steenrod operations) determined. That surely sounds conceptual, but try to find a conceptual proof! We have tons of explicit calculations in the classical and Novikov Adams spectral sequences, without which the solution of the Kervaire invariant problem would be unthinkable. Chromatic theory as a whole is informed by explicit calculations. In unstable homotopy theory, exponent theorems for the homotopy groups of spheres use a remarkable blend of theoretic and calculational techniques. There are tons of explicit calculations in ordinary and generalized cohomology in the past half century. Spin cobordism and its applications to curvature questions is an example of a blend of algebraic topology and differential geometry. I could go on for 100s of pages without pausing for breath, and none of these results use any model category theory, let alone $\infty$ categories. There is a subject out there, with real content.