Yes, that's how de Blasio phrases it, but I think there's more to it than how he's talked about it.
The way I see these mandates is an effort to have people maintain a person R-naught of less than 1. How each person achieve that is up to them. They can wear masks, they can get vaccinated, they can avoid places with high transmission rates (such as indoor dining/shopping), or a combination of all these things. And ideally we find more options for people to take so that that everyone can help achieve a personal R-naught in a way that best fits their needs.
Ultimately, the goal is to create a win-win scenario where on a personal level individuals can choose what degree of transmission prevention they want to pursue, and on a social level we're organized so that everyone is (hopefully) not infecting others, or doing so so sparsely that the infection rate is going down.
The goal now is to find a comfortable equilibrium in which society can continue to do its thing without explosive outbreaks of Covid. These sort of mandates help get us there. This is all still very new and we need to find more methods in which we can reach that equilibrium, and we should continue to look for them, but we can't let Covid-19 run rampant until we feel we've all the options we like.