We establish the first unconditional well-posedness result for the master equation associated with a general class of mean field games of controls. Our analysis covers games with displacement monotone or Lasry--Lions monotone data, as well as those with a small time horizon. By unconditional, we mean that all assumptions are imposed solely at the level of the Lagrangian and the terminal cost. In particular, we do not require any a priori regularity or structural assumptions on the additional fixed-point mappings arising from the control interactions; instead we show that these fixed-point mappings are well-behaved as a consequence of the regularity and the monotonicity of the data. Our approach is bottom-up in nature, unlike most previous results which rely on a generalized method of characteristics. In particular, we build a classical solution of the master equation by showing that the solutions of the corresponding $N$-player Nash systems are compact, in an appropriate sense, and that their subsequential limit points must be solutions to the master equation. Compactness is obtained via uniform-in-$N$ decay estimates for derivatives of the $N$-player value functions. The underlying games are driven by non-degenerate idiosyncratic Brownian noise, and our results allow for the presence of common noise with constant intensity.