Aims: We address two issues for the adoption of convex optimization in radio interferometric imaging. First, a method for a fine resolution setup is proposed which scales naturally in terms of memory usage and reconstruction speed. Second, a new tool to localize a region of uncertainty is developed, paving the way for quantitative imaging in radio interferometry. Methods: The classical $\ell_1$ penalty is used to turn the inverse problem into a sparsity-promoting optimization. For efficient implementation, the so-called Frank-Wolfe algorithm is used together with a \textit{polyatomic} refinement. The algorithm naturally produces sparse images at each iteration, leveraged to reduce memory and computational requirements. In that regard, PolyCLEAN reproduces the numerical behavior of CLEAN while guaranteeing that it solves the minimization problem of interest. Additionally, we introduce the concept of the \textit{dual certificate image}, which appears as a numerical byproduct of the Frank-Wolfe algorithm. This image is proposed as a tool for uncertainty quantification on the location of the recovered sources. Results: PolyCLEAN demonstrates good scalability performance, in particular for fine-resolution grids. On simulations, the Python-based implementation is competitive with the fast numerically-optimized CLEAN solver. This acceleration does not affect image reconstruction quality: PolyCLEAN images are consistent with CLEAN-obtained ones for both point sources and diffuse emission recovery. We also highlight PolyCLEAN reconstruction capabilities on observed radio measurements. Conclusions: PolyCLEAN can be considered as an alternative to CLEAN in the radio interferometric imaging pipeline, as it enables the use of Bayesian priors without impacting the scalability and numerical performance of the imaging method.