Like I said earlier - Vetinari has other means (besides the Watch) to deal with city matters so will be interested to see what angle you're coming at this now.
However - Carrot resigned
. Whatever Angua put in the note it was something that very clearly hit him very hard indeed and sent him running to Vetinari to burn his bridges and leaving Fred in charge. I cannot (and neither should anyone else) seriously think that Vetinari, faced with a complete clot in charge of the Watch and therefore pre-warned didn't have a back-up tactic to bring into play. That's just silly. He has spies everywhere and spies on the spies.
The city dwarves are artisans for the most part and so they will at least have custom/trade with the more reputable of the city's trades and 'arrangements' with the less savoury (the Thieves for instance). The Watch is important but only for the big stuff - the other minutiae of running the city is no less assessed, understood and eventually manipulated by Vetinari whose unofficial motto is 'if it ain't broke don't fix it'.
Of course he has other ways to apply pressure on fractious dwarves that will have far more immediate effects on them than what's happening out in Uberwald which will eventually settle one way or another. We see what Terry want us to see of A-M - Vetinari is the brain remember. He doesn't
have to act through the Watch in the City as such else why would he send Vimes away (and Cheery, Detritus and Angua by accident rather than his original design) when the urban dwarves were up in arms? The murders had already happened
and he still
sent the diplomatic delegation of his choice off to the source
of the discontent in Uberwald.
Carrot's resignation was an unforeseen surprise maybe (he did try to talk him out of it a little bit), but everything he'd set in motion (identified and helped by his finger on the pulse of Uberwald Lady Margolotta) was still in place with that one exception (Carrot) which he didn't
take steps to prevent in some way. In other words he had one less piece to manipulate at home but he'd have had other options (and that would be plural options) without him being in the City helping to keep the dwarves at home in order. Remember Vetinari is the Tyrant that got left to rot in his own dungeon G!G! and still
carried on manipulating and orchestrating Vimes in the defence of the city against the dragon. He has setbacks but they're usually
ones he's anticipated. Carrot resigning was unexpected but nevertheless still a blip because he didn't have to interfere with the Watch, simply because Fred kept them all from fouling up other skeins of the grand plan by keeping the rest of the Watch amused with the sugar thief hunt.
Finally Carrot the dwarf. He was brought up a dwarf and that's how he's 'cast'. That is his core sense of personal
. His core sense of important
lies in being a policeman because he will put the city first before dwarvish interests, if it puts the city in danger. He was Vetinari's first line of defence for the Watch
whilst Vimes was out in Uberwald but he wasn't
the only one.