This is an interesting question. I think that the correct answer is that the electron being fundamental is consistent with every experiment that we know.
I think I can do a bit better, though. Typically, theorists are pretty good at the art of dimensional analysis. Basically, every problem sets a typical scale for itself. The best example is quantum electrodynamics. It has only one scale---the mass of the electron. Because E = mc^2, this sets an energy where QED effects become important. If you work throught the numbers, you find that this scale is the Compton wavelength of the electron (NOT a coincidence). So you have to start worrying about calculating QED effects at the Compton wavelength of the electron, which is 10 or so times smaller than the Bohr radius. This is exactly why you can use old-fashioned non-relativistic quantum mechanics and get good results out of the Bohr model.
So, in some sense, the most natural place for new physics to take over is at a energy that the problem itself dictates. So if the electron was composite, we would have expected to find out that it was composite when doing experiments at an energy set by the electron's mass. And we've been doing those experiments for quite some time.
The same is true for the proton---when we start doing experiments at energies on the order of the proton's mass, we find out very quickly that the proton is composite.
So saying that the elctron is fundamental is a pretty good guess---there would have to be some pretty tricky new physics that we have missed. But, of course, anything IS possible.
