QCD is very well accepted, and in a way it answers your question, because from lattice-QCD simulations it is clear that QCD shows confinement, i.e., observables are always color-neutral states, and the hadronic spectrum (i.e., the masses of all the hadrons listed in the particle data book and even predictions for some not (yet?) observed) comes out pretty well. Together with also the successes of perturbative QCD, showing asymptotic freedom, explaining Bjorken scaling and its breaking, etc. also very well, this makes physicists pretty confident about the correctness of QCD, at least on the energy scales available in today's acclerators (particularly the LHC with the so far highest available beam energies).
Another question is, why the charge pattern of the elementary particles, i.e., the quarks, leptons, the gauge bosons, and the Higgs boson(s), is as it is. On the first glance one would think, it's totally arbitrary and just taken from observations in nature, as all the observable parameters like coupling constants have to be taken from experiment. At the second glance there is a restriction, because the weak interaction is based on a socalled chiral local gauge symmetry, which must never ever be explicitly broken by anything in the formalism, and one danger with the particular gauge group of the electroweak part of the standard model is that it could be explicitly broken by a socalled anomaly.
A theory has a symmetry, if the equations of motion of the dynamical degrees of freedom, which are in this case all the fields, describing the elementary particles and their interactions, do not change their form when one does some non-trivial transformation. The electroweak standard model is constructed on the classical level such that it obeys this chiral local gauge symmetry, which means (loosely speaking) a symmetry under rotations of all the matter fields in a particular way, and these rotations can even depend on space and time variables. This is a pretty strong constraint on the theory, but it leaves also enough freedom to implement all the so far observed facts about the particles and their weak and electromagnetic interactions.
Now this classical field theory is not directly useful to describe high-energy collisions. Only the electromagnetic interaction has a very well-known classical limit, namely classical electrodynamics, which governs all our electronic devices around us. For the high-energy particle physics, however, one must use it as the starting point to build a quantum field theory. A quantum field theory is just the quantum theory of many particles, which are described as specific excitations of the quantum fields, that can also describe the creation and annihilation of particles, which is exactly what's happening in collisions at high (relativistic) energies. Now there's a mathematical recipe to make a classical field theory a quantum field theory, and you can derive the famous Feynman rules of perturbation theory from it, which tells you how to predict cross sections for scattering processes.
Now there's the danger that the symmetry, the classical field theory obeys by construction, gets lost in the quantization process. This is called an anomaly. If this happens to a local gauge symmetry, it's a desaster, because then the theory produces totally non-sensical results, where the probabilities for all possible scattering processes are not adding up to 1 as it should be, or they become even negative. So one has to avoid this desaster of anomalous breaking of the local gauge symmetry. The mathematics behind it, group theory, tells us, when such symmetries can occur. Some groups are anomaly safe to begin with. Unfortunately the electroweak gauge group is not such a nice one, but it can potentially be anomalously broken just by quantizing it. Another theorem, however, tells us, how to efficiently check, whether this desaster really occurs. One has to calculate only a specific set of one-loop diagrams. This calculation shows that one can perhaps avoid the anomaly by choosing the right pattern of charge values of particles, and amazingly, the charge pattern of the standard model is precisely such that the anomaly is cancelled! For me that's one of the most beautiful outcomes of theoretical physics! Indeed it happens that just because each flavor family has one lepton with a charge -e, its neutral neutrino (and the corresponding anti-particles) and two quarks, one with -1/3 e and one with 2/3 e charges coming in three "copies" (i.e., each quark carries also three color charges, which is the coupling of the strong force in the strong sector of the Standard Model, QCD) and the corresponding anti-quarks with the opposite charges and anti-colors. This charge pattern is such that the anomaly cancelled, and the Standard Model is thus free of anomalies of its fundamental local gauge symmetries. One should however say, that the particular charge pattern is not the only one that prevents the anomalous breaking of the electroweak theory. So there remains still the question, why nature has chosen the observed charge pattern of elementary particles and not just another one.
One should mention that sometimes anomalies are also good! E.g., in the Standard Model there are also accidental symmetries. The most important example is the approximate chiral symmetry of the light-quark sector of QCD. This allows one to build effective hadronic models, i.e., to describe hadrons, which are composite objects made of quarks and gluons, as elementary particles at low energies, and the residual strong interactions among the hadrons is ruled by this approximate chiral symmetry. On the other hand, chiral symmetry is also a problem, because it predicts the decay rate for the process ##\pi^0 \rightarrow 2 \gamma## (the decay of a neutral pion to two photons) way too low. However, the pion is a pseudoscalar particle, and there's a particular anomaly which leads to the correct value for this decay rate. Here, nothing is destroyed by the anomaly, because it violates a symmetry of the classical theory which is unimportant for the mathematical consistency and physical interpretabilty of the corresponding quantum theory.
