Vimes’ objections aren’t about the degree of political power. They’re about the very nature of Kingship. The Patrician is in effect a monarch in the literal sense of sole ruler. But he’s not a King because he’s not set apart as somehow divinely or magically suited to rule.
Carrot of course is the heir to the throne, and does have this aura of specialness, which is why he agrees he should be a humble Watchman and not on any kind of pedestal.
“No more kings. Vimes had difficulty in articulating why this should be so, why the concept resonated in his very bones. After all, a good many of the patricians had been as bad as any king. But they were…sort of…bad on equal terms. What set Vimes’s teeth on edge was the idea that kings were a different kind of human being. A higher lifeform. Somehow magical.”
― Terry Pratchett, Feet of Clay
“The King doesn’t have any power, he’s just this really special guy everyone looks up to” - this is anathema to Vimes.
“Royalty was like dandelions. No matter how many heads you chopped off, the roots were still there underground, waiting to spring up again.
It seemed to be a chronic disease. It was as if even the most intelligent person had this little blank spot in their heads where someone had written: “Kings. What a good idea.” Whoever had created humanity had left in a major design flaw. It was its tendency to bend at the knees.”
But what if it was Carrot. Vimes doesn’t just like Carrot, he knows he’s a good man, right?
‘Listen! We probably had good kings, once! But kings breed other kings! And blood tells, and you end up with a bunch of arrogant, murdering bastards! Chopping off queens’ heads and fighting their cousins every five minutes! And we had centuries of that! And then one day a man said 'No more kings!’ and we rose up and we fought the bloody nobles and we dragged the king off his throne and we dragged him into Sator Square and we chopped his bloody head off! Job well done!’
'Wow,’ said Carrot. 'Who was he?’
'Who?’
'The man who said 'No more kings.“
People were staring. Vimes’s face went from the red of anger to the red of embarrassment. There was little difference in the shading, however.
'Oh… he was Commander of the City Guard in those days,’ he mumbled. 'They called him Old Stoneface.’on Old Stoneface | Terry Pratchett, Men at Arms
Note this is specifically about the role of King. A triumvirate with Moist as the agent of change, Vimes keeping things stable and Carrot as the “face” would work well. But Vimes and Kings are incompatible.
Pratchett moved Ankh-Morpork from medieval city-state with the Patrician and Guilds to a quasi-Victorian industrial city with corporate interests, a solid financial system, newspapers and written communications ( implying a literate society), and a population capable of activism in pursuit of sub-groups (dwarves, golems) rights and interests. It’s not hard to see something quasi-democratic arising out of that. There is certainly going to have to be a means for all these interests to have a voice and to have their point of view taken seriously (at least to keep people off the streets) and a means for balancing all of these in pursuit of some broadly acknowledged all-Ankh-Morpork goals.
The council already exists; you can see that elections (corrupt ones, obviously, this is A-M) for the role of council representative could easily emerge within one or two communities. You could even see the emergence of new communities defined along different lines - class, perhaps. Where is the voice for the common people we saw in Unseen Academicals?
The Patrician has essentially been holding the political development of Ankh-Morpork back by being so good at his job. In his absence, a lot of stuff is going to happen very quickly and when the dust settles OMOV will be over.