You are well out of date with your terms. Tory meant supporting the Stuarts around the time of the Glorious revolution. Not the best part of a century afterwards. As I mentioned, there were several dozen MPs openly describing themselves as Tories at this time, and that would have been politically inpossible had it been a synonym for Jacobite.
As for your point about winning elections, that is extremely misleading. The reality is the King wouldn't go to the country for an election unless he was sure of victory. He only did it when he was confident, and when he wasn't he tried to get the administration most favourable to him from the current parliament. You mention the Fox-North coalition. George III's hatred of Fox was well known, so why would he have included him in government if he could have just appointed North and then won an immediate election?
It is also well known that by William IV's time, the King had virtually no power at all. So if your claim of utter royal control until the early 1800s was true, there would have been a complete turnaround in just a couple decades without any major event happening. Clearly that is unbelievable. In fact royal power had been declining in continuous fashion since 1688.
There were Tories, but until the acession of George III, every man in power and every man who even hoped to be appointed to office, called themselves Whigs. Even Lord North and Bute called themselves Whigs. Only their opponents called them Tories.
The label 'Tory' was, in this sense, applied to the Prime Ministers Lord Bute (1762–1763) and Lord North (1770–1782); but these politicians considered themselves Whigs. In his study of the debates in Parliament for 1768-1774, P. D. G. Thomas discovered that not a single politician labelled themselves a Tory.
The reason why the king would win elections, is because he controls patronage through the Treasury, and he controls the rotten boroughs, etc., and a majority of MPs were susceptible to such shenanigans, thus, any ministry he appoints is assured of winning any election. Even when is extremely unpopular, like in 1784 after losing the American War, he still won overwhelmingly over the more popular Fox in supporting Pitt the Younger. Even in 1780, with an unpopular American War, he still won the 1780 election for Parliament for Lord North.
Sure, the king could be compelled to accept ministries he didn't like. But he could easily dismiss them too. How many ministers that he hated really lasted? The ones that had really long terms were Lord North and Pitt the Younger, both of whom were George III's men through and through.
But ask yourself this, if George isn't powerful or in charge, how can he appoint Pitt the Younger in face of overwhelming Parliamentary Hostility in 1783? How many votes of no confidence did Pitt and George III ignore during that time? Then at the height of Pitt's popularity and power, dismiss him in 1801 and replace him with Addington when Pitt dared to disagree with George III over the Catholic Emancipation? Then dismiss the Ministry of all the Talents in 1807 when they proposed Catholic emancipation after only a year on power? Fox in 1783, Pitt in 1801, and the Talents Ministries were popular, and were supported by Parliament. Once the king dismissed them, that support evaporated, and went to the ones he supported and appointed in their place, in this case Pitt the Younger in 1783, Addington in 1801, and Portland in 1807.
That to me tells me that George III is quite a powerful monarch, right up to 1811 when he became insane. But before that, he was in charge. Was he as powerful as James II? No, but he not a cipher, and more often than not, the arbiter of the executive branch of the British Government right up until his permanent insanity in 1811.