If the second and third matter families (so-called generations: muon, muon neutrino, tau, tau neutrino, ...) didn't exist, would that affect how the universe runs? Are they optional or required?
This question can be split into two parts: How do three generations affect the way the universe runs within the standard model, and how do they affect beyond standard model physics?
It is likely that the three generations are important in beyond standard model physics and without them particle physics would be very different. However we do not have any definite idea about how that works so let's concentrate on standard model physics.
It is true that almost all physics that affects the way the universe runs depends on only the first generation of matter particles, i.e. the up and down quarks, electron and electron neutrino, plus the gauge bosons. Although we can now easily observe particles from the second generation i.e muons, strange particles and charm, these have almost no significance in chemistry or even main processes of nuclear physics. Hence their effect on how the universe runs is very small. The third generation has even less influence.
However there are some exceptions and possible exceptions.
(1) Muons are an important component of cosmic rays and it is now thought that cosmic rays can play an important part in weather and climate. There have been studies to observe correlations between muon flux and upper atmosphere temperature. Even if this is observed it does not necessarily mean that muons have a causal affect on weather but it is possible. Atmospheric muons could have other important influences such as gene mutation, but this is speculative as far as I know.
(2) CP violation can occur in the standard model at observable rates only through the CKM and PMNS mixing matrices for three generations. With only one or two generations CP violation in the standard model can only happen via unobservable non-perturbative effects. While CP violation does not have any affect on the way the universe runs now, it is thought to be essential in big bang cosmology in order to create an imbalance between matter and anti-matter. Without it all matter would have annihilated with antimatter as the universe cooled. However, it is not known if other CP violating processes from beyond standard model physics that do not depend on the three generations are more important.
(3) In calculations of particle masses and decay rates the existence of second generation particles as virtual particles has some effect. You could argue that the masses of protons and neutron might be slightly different etc if there were no muons etc.
(4) The three generations of neutrino have observable effects on decay rates of weak gauge bosons and also on cosmological parameters measured in the cosmic microwave background. Whether this affects the running of the universe in some significant way is another question.
(5) The top quark mass is an important factor affecting the stability of the vacuum and the allowable mass range of the Higgs boson for stability. You could therefore argue that the top quark and perhaps some of the other heavy flavour particles are needed to stop the vacuum decaying.
In several of these possibilities you could make the point that without the second and third generation the parameters of the standard model could be adjusted to produce the same effect, so it is not certain that the second and third generations have any real necessity in the running of the universe.
I guess protons need strange quarks. eg.
The a particle later to be known as the electron neutrino was proposed to exist by Wolfgang Pauli because physicists already knew then that it must exist--otherwise the known laws of conservation of energy, momentum and angular momentum would be violated. They were studying beta decay of neutrons: $n^0 → p^+ + e^−$, and found that while the charges conserved, the energy and momentums did not. A hypothetical particle was proposed to fill in these gaps (this is why scientific laws are so useful in that they have predictive quality). It was subsequently shown that just one particle which filled in all the gaps of beta decay existed--the electron neutrino (Cowan, Reines, 1956).
With the discovery of the electron neutrino, physicists were puzzled by how few were reaching earth from the sun's fusion reactions. From known values of solar radiation and knowledge of fusion mechanisms including beta decay, we were detecting way too few electron neutrinos--a big hole in our understanding (based on the Standard Solar Model) later known as the solar neutrino problem. It was again, another implied violation of conservation laws. A year later Fermi's assistant proposed that neutrinos change into different flavours (muon, tau neutrinos) just like quarks were known to do. It was just a matter of time that experimental physicists found the next generations of the neutrino.
As you may know this year's Nobel in physics was awarded to the people who hypothesized the Higgs boson, which had to exist in order to explain the masses of the W and Z bosons and in a way which did not conflict with gauge theory which was by then too successful to abandon.
To answer your question, yes there is a theoretical need for three generations of leptons to exist mostly due to the fact that the foundations of physics would collapse and everything else would follow. Instead we assume that everything does not collapse, that everything works, and we just have to discover what those things are that make everything work the way it works.