I've been wondering about this for a while too.
In the direction of a topological Jacquet–Langlands correspondence, recently we have [this paper][1].
You will also find some pitfalls in this theory.
For example, if you believe that the Lubin–Tate tower should in some sense incarnate a local Langlands correspondence, the fact it does not naïvely descent to the sphere is a problem.
To my knowledge, Andrew Salch (the auther of the paper I linked) is the person who thought about this type of topological Langlands correspondence the most, I can recommend also reading his other papers.

You mention you recently started learning about homotopy theory, so maybe this recommendation is too advanced, but yes, there is a theory of spectral abelian varieties and also a theory of moduli of spectral elliptic curves. For this you should read Lurie's series of papers on elliptic cohomology.

There is also a notion of topological automorphic forms, however to my knowledge it is not clear, what should happen on the Galois side.
Classically one would consider representations of the Galois group of $\mathbb{Q}$ (to be more precise this only sees cohomological automorphic forms, but to my knowledge topological automorphic forms so far also only generalize cohomological automorphic forms), the rationalization of the sphere is just $\mathbb{Q}$ however (and in any case the étale site is nil-invariant and cannot see the difference between the sphere and $\mathbb{Z}$, so one has to do something smarter than étale cohomology).
Even with recent advances in categorical Langlands theory none of the proposed candidates on the Galois side that I know of seem to make sense or yield something new when considered over the sphere.

Also note that spectral algebraic geometry is a very different beast from derived algebraic geometry, and so far it seems to me that advances in elliptic cohomology yield applications to chromatic homotopy theory. So far I have not seen an application to number theory.


  [1]: https://asalch.wayne.edu/level-structures-via-BG-3d.pdf "Salch - \$\ell\$-adic topological Jacquet–Langlands duality"