Here is another heuristic, related to what Randal-Williams said above. The sphere spectrum is the unit object in nice categories of spectra. That is, ring spectra are algebras over the sphere spectrum. Now, to every scheme X you can associate a K-theory ring spectrum K(X), and this is contravariant. So, in the usual theory there is a morphism K(Z)->K(X) for all schemes X. So, finding F_1 also means finding something (it's K-theory spectrum) that maps to the (homotopy) limit of all K-theory spectra. That this should be the unit object of the category of spectra doesn't seem very surprising.

Here is another heuristic, related to what Randal-Williams said above. The sphere spectrum is the unit object in nice categories of spectra. That is, ring spectra are algebras over the sphere spectrum. Now, to every scheme X you can associate a K-theory ring spectrum K(X), and this is contravariant. So, in the usual theory there is a morphism K(Z)->K(X) for all schemes X. So, finding F_1 also means finding something (its K-theory spectrum) that maps to the (homotopy) limit of all K-theory spectra. That this should be the unit object of the category of spectra doesn't seem very surprising.