Why was the $Z$-boson first theorized? What problem did it solve? I have looked around, but I can't find a single (or simple-ish) explanation as to why the $Z$-boson was first dreamed up by theorists, years before it was discovered.
 A: Probably an historical point of view is the best way to explore you question. At the time physicists were trying to unify QED, and so elecromagnetism, with what they new about weak interactions, which were explained by Fermi theory.
Let's begin, as we always must, to find the symmetry group of the theory. At the time, they knew that there must be one gauge boson for the photon. Moreover there must be another two vector bosons for the $W^\pm$ fields. The existence of the $W^\pm$ was theorized to explain the decay of the muon and the neutron.
With this we need at least the $SU(2)$ symmetry group since it has $3$ generators. But it turns out that this group is too small since it only accounts for left-handed interactions and we know that electromagnetism is perfectly symmetric between left and right-handed fermions.
What Glashow proposed was the following minimal group
\begin{equation}
    SU(2)_W\otimes U(1)_Y
\end{equation}
where the reps are defined by the isospin symmetry and the hypercharge. The $U(1)_Y$ is not to be confused with the $U(1)$ symmetry group of electromagnetism, that will come later after symmetry breaking.
Based on this symmetry group, the existence of a fourth gauge boson was theorized since the group has $4$ generators. It will turn out that the additional gauge boson is, in fact, the $Z^0$ which mediates the weak neutral currents.
The EW theory based on this minimal group, the Glashow-Weinberg-Salam (GWS) theory, explained really well all the experiments when coupled with the theory of quark mixing first theorized by Cabibbo and then introduced with the Cabibbo-Kobayashi-Maskawa (CKM) matrix.
The GWS theory was proven to reconstruct, at low energies, the Fermi theory of weak interactions which worked quite well at the time.
The only problem was the prediction of the additional gauge boson which could mediate weak neutral decays. It's discovery proved that indeed the minimal group proposed by Glashow should be the right symmetry group for the EW unification.
A: 
Predicting the W and Z


Following the success of quantum electrodynamics in the 1950s, attempts were undertaken to formulate a similar theory of the weak nuclear force. This culminated around 1968 in a unified theory of electromagnetism and weak interactions by Sheldon Glashow, Steven Weinberg, and Abdus Salam, for which they shared the 1979 Nobel Prize in Physics. Their electroweak theory postulated not only the W
bosons necessary to explain beta decay, but also a new Z  boson that had never been observed.

So trying to find a similar theory to quantum electrodynamics for the weak interaction led to a theory that had a Z Boson in its postulates.
Italic mine.

The fact that the W and  Z bosons have mass while photons are massless was a major obstacle in developing electroweak theory. These particles are accurately described by an SU(2) gauge theory, but the bosons in a gauge theory must be massless. As a case in point, the photon is massless because electromagnetism is described by a U(1) gauge theory. Some mechanism is required to break the SU(2) symmetry, giving mass to the W
and Z in the process. The Higgs mechanism, first put forward by the 1964 PRL symmetry breaking papers, fulfills this role

The theory developed slowly over that period during the search for  a unified theory of weak and electromagnetic interactions.
