It's nice you have acknowledged that Manchester City has been performing at the top for several seasons in recent years, which was also where I concentrated on. In this same recent years, Liverpool has won 2 EPL titles and one UCL while Manchester City has 6 EPL and one Champions league, in the same recent years Chelsea has 1 EPL, one UCL and one CWC of which they just won few weeks ago. By contrast among these teams, Manchester City still has more in the last ten years. So if any club fans should be noisy, it should be Manchester City fans.
When these banters and the noise are made between fans of different clubs I am sure we all know that each club fans do hit hard in the discussion on the league competition which they have greater accomplishments than the other club. For example, Manchester city in the past 10 years may have won the EPL trophy more than any team in the EPL but in European league competition like the UCL they don't really have much to boast of like Chelsea fans will do. Same goes for Manchester United, Liverpool, each of these clubs has some areas of weakness and strength in all competitions. But in all of this, the fan base matters a lot in determining the noise too.
Greater accomplishment you say, by this you're talking of total trophy in each of the competition. Even if you look at this, Chelsea still has no mouth in any of them because there's no trophy they have more than any other team in the league. If you talk of UCL they have only two, Liverpool has 6, Manchester United has 3 and Manchester has one, Arsenal non. In this category Liverpool and Manchester United fans should be making the noise here. If you talk about the league, Liverpool has 20, Manchester United has 20, Arsenal has 13, Manchester City has 10 and Chelsea 7, they are still last in this category and they still make the most noise. In recent years of winning the league, it's still Manchester City that has the most 6. If you talk about FA, Arsenal has the most seconded by Manchester United. You see, in these categories, they have teams that are ahead of them.