I'm seriously confused here--are you sure the car was "on" rather than in accessory mode?
In Accessory mode, everything is running off the 12V battery, and, as in any car with a key it will die pretty quickly.
But in regular "On" mode, I've checked twice and the car was
definitely keeping the 12V battery floated using the high voltage battery pack. I would expect that, in this state, when the high voltage battery pack gets low the car will start the
ICE to keep it from discharging too far. That, I admittedly haven't tested, but I
have had the car turn the ICE on while sitting turned on in park for 10 minutes or so even when the high voltage battery had plenty of energy left...
I understand shutting down after too long in park (Volts now do this after 90 minutes because people early on forgot to turn them off and the garage filled with exhaust gas because the car kept cycling the ICE). The manual doesn't say one way or the other anywhere I can find, but I'd expect it to either shut itself off after a while or just keep cycling the ICE. Running down the high voltage battery pack then letting the 12V battery die seems like a bizarre misfeature.
One thing to note is that you're probably not supposed to jump start a regular car (ice vehicle) with the Clarity or any other hybrid. It's very likely the 12 v battery is not a high current battery.
Due to the capacity of the DC/DC converter that feeds the 12V battery off of the high voltage battery pack, the Clarity should have absolutely no trouble jumping another car. I've never tried it with my Clarity, but I jumped a few vehicles with a Volt without issue (I know that DC/DC converter is rated at around 60A before you even start drawing down the 12V battery, and the Clarity should be similar given how many high-power climate control things run off 12V).
Conversely, since it's not the 12V battery starting the ICE, it should in theory be extremely easy to jump a Clarity with a 12V battery--you just need to supply enough power to get the electronics to power up long enough to get the DC/DC converter going. Even one of those cheesy "pocket jump start" things that don't actually work on regular cars should be fine.
Looking into "eye" of 12v batt charge indicator on top of the 12v battery... looked normal charged colors.
I could be mistaken, but per page 508 of the manual I don't think that indicator has anything to do with state of charge--I think it has to do with electrolyte (and therefore battery) health.