The King's Quarry: How Louis XVI Went from Hunter to Hunted

Yes, the flight to Varenne was the first crack between Louis XVI and the national assembly, but constitutional monarchy with him at the head was still the preference amongst most of the assembly (the first constitution confirm that).

What did him was that he declared an unjustified war on his in-laws, starting an Alliance game that put France alone against 6 countries and kept vetoing any mobilisation order. Les Tuileries got invaded by rogues elements, and letters that prove that the king was selling the army (describing future troop movement to the Austrians) were found. That treason basically started the Terror, and pushed the assembly to vote on declaring the republic (and in the end killing the king. It was a 60/40 vote if I remember correctly)