Canada does have a sort-of similar system. It's just that the "president" in Canada is "the crown", which is the Governor General representing the current British monarch. It's much more of a ceremonial role in Canada, but technically the Governor General does appoint the Prime Minister.

Australia has essentially the same system as Canada. In 1975 the Australian Governor General dismissed the Prime Minister and picked the leader of the opposition as Prime Minister so that he could call an election. Described like that it seems like a blatant abuse of power. But, the background was a really dysfunctional government. One party had narrow control over the house, the other had narrow control over the senate, and the senate was blocking everything the house tried to do. I don't know the full details of what happened in that affair, but it seems like it could be a good thing if a Governor General would step in in a crisis resolve a deadlock.

Canada also has the "confidence votes" part of the crisis in France. AFAIK in Canada losing a confidence vote immediately triggers an election, unlike in France where it can just lead to a scramble to see who can become the new PM among the existing representatives. Because triggering an election is a big deal, it doesn't tend to happen too often. But it has happened. In 2011 Stephen Harper's government lost a confidence vote, and there was an immediate election, but he won that election. In 2007 Paul Martin's government also fell to a confidence vote.