But no, the tau isn’t made up of anything, at least not according to all the observations that we currently have. There are several reasons physicists know this.
First, if the tau was made up of those other particles, you’d have to find a way to hold them together. This would require a new force. But we have no evidence for such a force. . . .
Second, even if you’d come up with a new force, that wouldn’t help you because the tau can decay in many different ways. Instead of decaying into an electron, a tau-neutrino and an electron anti-neutrino, it could for example decay into a muon, a tau-neutrino and a muon anti-neutrino. Or it could decay into a tau-neutrino and a pion. The pion is made up of two quarks. Or it could decay into a tau-neutrino and a rho. The rho is also made up of two quarks, but different ones than the pion. And there are many other possible decay channels for the tau.
So if you’d want the tau to be made up of the particles it decays into, at the very least there’d have to be different tau particles, depending on what they’re made up of. But we know that that this can’t be. The taus are exactly identical. We know this because if they weren’t, they’d themselves be produced in larger numbers in particle collisions than we observe. The idea that there are different versions of taus is therefore just incompatible with observation. . . .
A decay is really just a type of interaction. This also means that all these decays in principle can happen in different orders. Let’s stick with the tau because you’ve already made friends with it. That the tau can decay into the two neutrinos and an electron just means that those four particles interact. They actually interact through another particle, with is one of the vector bosons of the weak interaction. But this isn’t so important. Important is that this interaction could happen in other orders. If an electron with high enough energy runs into a tau neutrino, that could for example produce a tau and an electron neutrino. In that case what would you think any of those particles are “made of”? This idea just doesn’t make any sense if you look at all the processes that we know of that taus are involved in.
Everything that I just told you about the tau works similarly for all of the other unstable particles in the standard model. So the brief answer to the question why elementary particles can decay is that decay doesn’t mean the decay products must’ve been in the original particle. A decay’s just a particular type of interaction. And we’ve no observations that’d indicate elementary particles are made up of something else; they have no substructure. That’s why we call them elementary.