This is an alright use of a monarch IMO.
The UK example would be if the UK government lost a vote of confidence, the monarch might ask the opposition parties to see if they could form a government that could pass a vote of confidence to avoid a general election. Although parliamentary math makes such a scenario unlikely in practice.
On the UK end I don't think the monarch can sack the Prime Minister. While the UK monarch could strongly suggest to a Prime Minister that it's time to resign. Usually when a Prime Minister gets to that point, it's usually the end of the road and they won't last much longer anyway even if they pushed forward.
The monarch has a lot of soft power which under the modern system is usually deployed to ensure continuity of government rather than in a controlling manner.
(The Australia, Canada, etc constitutions gives the monarch different powers than the UK and it's usually delegated to whoever is the governor for that nation)