Why does the $1/2$ spin of fundamental fermions (electrons, quarks, and neutrinos) split them into three variants that differ only in mass, while the integer spins of massless fundamental bosons (e.g. photons and gluons) causes no such splitting?
If the bosons had mass, would they also split into three generations?
The number of gauge bosons is restricted by symmetry: a given theory with a certain gauge invariance admits as many gauge bosons as there are generators of the corresponding gauge group. For example, there is one generator for $\mathrm{U}(1)$, resulting in the existence of a photon. $\mathrm{SU}(3)$ admits eight generators, which yield eight gluons. This is true without any reference to generations. Regarding the Higgs, it is not entirely clear that there exists only one particle.
The number of fermionic (matter) particles in the Standard Model is not dictated by gauge symmetry and in principle one could construct a model with more or fewer generations. The question as to why there are exactly three generations is still an open problem, even in string theory.
There are two things that define a particle physics model (at low energies). The first one is the gauge group G we want the model to be symmetric under. For the Standard Model (SM) we set this to $G=SU(3)\times SU(2)\times U(1)$ (for good experimental reasons!). This will uniquely determine the number of gauge bosons needed to make the model consistent.
The second ingredient is the matter content, i.e. how many fermions and scalar fields we want the model to include. In the SM we choose 3 generations of fermions (this includes the leptons, quarks and their CP partners) and 1 scalar field (the Higgs).
From a low energy point of view, the two choices are completely independent and we can choose anything we like. For example it is conceivable to have a model with $SU(5)$ gauge symmetry and no matter at all, etc...
However, the OP is not alone in not being satisfied with this apparent lack of symmetry in treating fermions and bosons. Most physicists would indeed agree that we should try and find a more unified description and hope that nature has such a bigger symmetry at a deeper level. Theories beyond the Standard Model are indeed following such an approach. The best studied example are supersymmetric theories in which every boson/fermion has a superpartner that is a fermion/boson. In this theories, the symmetry of numbers is restored!
All in all, to answer questions like the number of generations in the SM, the masses of the fermions, etc you need a candidate for a complete theory at high energies and the answers will depend on this candidate. String theory is considered to be the most successful framework for this job and string models do indeed make concrete predictions about (among other things) the number of generations we should be observing. Unfortunately, there are too many models (vaccua) to choose from and no obvious way to make the choice...
