This manuscript is concerned with the Keller-Segel-Stokes system(⋆){nt+u⋅∇n=Δn−∇⋅(nF(|∇c|2)∇c),ct+u⋅∇c=Δc−c+n,ut=Δu+∇P+n∇Φ,∇⋅u=0, under no-flux/no-flux/Dirichlet boundary conditions in smoothly bounded three-dimensional domains, with given suitably regular functions F and Φ. Here in accordance with recent developments in the literature on refined modeling of chemotactic migration, the introduction of suitably decaying F is supposed to adequately account for saturation mechanisms that limit cross-diffusive fluxes near regions of large signal gradients. In the context of such nonlinearities which suitably generalize the prototype given by F(ξ)=KF(1+ξ)−α2, ξ≥0, with KF>0, known results addressing a fluid-free parabolic-elliptic simplification of (⋆) have identified the value αc=12 as critical with regard to the occurrence of blow-up in the sense that some exploding solutions can be found when α<12, whereas all suitably regular initial data give rise to global bounded solutions when α>12. The intention of the present study consists in making sure that the latter feature of blow-up prevention by suitably strong flux limitation persists also in the more complex framework of the fully coupled chemotaxis-fluid system (⋆). To achieve this, as a secondary objective of possibly independent interest the manuscript separately establishes some conditional bounds for corresponding fluid fields and taxis gradients in a fairly general setting that particularly includes the subsystem of (⋆) concerned with the evolution of (c,u,P). These estimates relate respective regularity features to certain integrability properties of associated forcing terms, as in the context of (⋆) essentially represented by the quantity n. The application of this tool to the specific problem under consideration thereafter facilitates the derivation of a result on global existence of bounded classical solutions to (⋆) for widely arbitrary initial data actually within the entire range α>12, and by means of an argument which appears to be significantly condensed when compared to reasonings pursued in previous works concerned with related problems.