Iterative methods such as Fixed-Point Iterations arising from the discretization of PDEs by standard methods (FE, FV…) offer their share of challenges when applied to large systems due to their poor conditioning. Slow convergence, limit cycle oscillations and, in the worst case, divergence are regularly observed. Deflation-based techniques such as the Recursive Projection Method (RPM), are particularly appealing for tackling this kind of issues without sacrificing the physical fidelity, as they allow to discriminate in between slow and fast modes, the first one being solved implicitly while the baseline algorithm is still relied upon to solve the latter. More recently, an alternative approach, called BoostConv, has emerged around the same ideas, although the connection between the two has remained silent. In this paper, we offer to comprehensively revisit this family of methods for fixed-point iterations, giving rise to a new unified framework that encompasses both approaches but also, apparently unrelated ones, such as Additive Correction Multigrid, and brings to light a number of as yet unexplored degrees of freedom, auguring further potential improvements in terms of performance and robustness. The present analysis is structured about 3 axes: choice of a projector, trouble modes recruitment, and non-linearity management. A comparison of existing approaches will be conducted, and the benefit of proposed new variants exemplified, on a selection of incompressible CFD cases with varying complexity, time-steps and mesh sizes, for sake of completeness.