Correct, the Governor General chooses the PM based on who is most likely to be able to lead Parliament. This is normally the leader of the majority party. If there is no majority party, the Governor has to make a decision.
Surely he asks the leader of the largest party if they are able to form a government, with a working majority or by coalition. If they decline or fail, he then asks the question of the next largest party. If they also fail, it is open to him to call another election.
The Existing Prime minister and cabinet remain in control till a replacement with a majority is found.
Prime minister in most commonwealth countries use this system. Prime ministers are not elected.
It is the Government that is in power not the Prime minister, they can be replaced with out a further election.
Sovereignty lays with Parliament not the Government.
Parliament can vote down a prime minister and government at any time.
Ultimate Sovereignty is held by the people, who elect the individual members of parliament.