The moment a government in a Westminster parliament loses a confidence vote, they become a caretaker government, a very constitutionally bounded creature. More importantly, their ability to advise the Sovereign/Governor General becomes extremely limited; they can't advise the GG to make new appointments, make most orders in council, or pretty much anything beyond keep basic organs of government going.
In a no confidence situation, it becomes the Governor General's job to figure out what to do next, and the government, being a caretaker, no longer can advise on the use of Royal Prerogatives such as dissolution or appointing new ministers (a new government).
A caretaker PM can certainly tell the GG what he thinks, but as happened in British Columbia in 2017, when the Premier of the province, having lost a confidence motion on the Throne Speech, tried to convince the Lieutenant-Governor to dissolve the legislature and call new elections, the vice-regal representative is under none of the obligations that a premier or PM who enjoys the confidence of Parliament has. In that case, the LG simply rejected the advice, and asked the opposition leader to form a government.
This is why the concept of confidence (and its loss) is far a better moderator of government excesses than the much older notion of impeachment. The latter evolved as Parliament in England gained more authority, but could not directly go after the King, so would often go after the King's ministers and agents through the use of impeachment. But even by the American revolution, impeachment in the Westminster constitutional order had fallen into disuse in preference to confidence. One of the first governments to fall to a loss of confidence was the Ministry of Lord North, after the defeat of the British in the War of Independence.
In general, I don't think someone of Trump's demeanor would ever be able to get away with as much in a Westminster government. Boris Johnson probably pushed the margins as much as any modern Prime Minister in the UK, and in the end he was effectively removed by his own party. It was an even swifter judgment for Liz Truss, who ended up serving the shortest amount of time as PM, beating George Canning, who died in office after 119 days in 1827.
Here in BC we've had multiple Premiers forced to resign. The closest analog to Trump was Bill Vander Zalm, who was accused of a serious conflict of interest over the sale of one his personal properties. He hung on for some time after the allegations became public, and while he ultimately resigned in disgrace, his cabinet was sufficiently worried that he might ignore all pleas to depart that they they hatched a scheme with the Lieutenant-Governor to have the government vote no confidence in itself, which would have forced Vander Zalm to resign, and then the Lieutenant-Governor would ask the designated member of cabinet to form a new government.
In short, in the Westminster system, the Sovereign and his representatives hold certain reserve powers that function as negative powers; almost never used, but the mere fact that they do not accessible by the government of the day creates a ceiling on the constitutional games that can be played. What's more, there are both visible ways to get rid of errant PMs and Premiers (leadership reviews, cabinet revolts, caucus revolts) and much quieter ones (ministers using their access to the King/GG/LG to get around a head of government).
The US put all its eggs in one basket by making a unified singular executive with powers commensurate with a Tudor-era monarch, the Westminster system created a split executive, with an Efficient part that does all the ruling, and a Dignified part that reigns.
