> Nobody has even close to an absolute majority in the parlement.

If the parliament would elect the government, the situation would be exactly as problematic as it is now. I don't see how the fact that the president chooses the prime-minister has any impact on the stability of the system.

You have pointed out a lot of true things, but none of them seems to be a your-system-specific-cause of the current instability.

I don't understand how a 30-30-30 split would work any better if parliament elected the government instead of the president. They would still be unable to reach any agreement, including an agreement on who to put in the government.