Push starting is just an old cure for several different ailments. Could be weak battery. Could be bad solenoid starter or bindix (sp?) gear. While it is great to live in the hopes of never having one of these breakdowns, I have lived through multiple events where push starting saved the day. True, there were no cell phones and that makes it easier to call for the local assistance and pay them to save you rather than your home mechanic (or Pep Boys in this case).
At such times, eroding your catalytic converter is the last thing on your mind. For the most part, I do not see how a manual transmission vehicle could be seriously hurt but one or two push starts and I would expect that it would work.
Automatic transmissions are fairly different. Some work and some do not. The last time I push started (hill roll started really) was a in a 1960s era T-bird (I want to say 1963, but not sure). Yes, it was old then but it got me around. Anyway, I do not know of any new ones that do; but that does not mean that there are not any.
I was not meaning for this to be a thread on "Should you push start your Elio every morning to avoid maintaining it?" but rather, in the manual transmission case, "Is there any fundamental engineering issue which will likely cause immediate failure if I push start the manual transmission Elio?"