For certain initial value problems, physically admissible solutions should remain within a convex set. For instance, with the Navier-Stokes and compressible Euler equations, density and pressure must remain positive at each time step. However, positivity is not guaranteed in scale-resolving simulations, particularly when using high-order discontinuous spectral element methods such as flux reconstruction schemes. On the other hand, when positivity-preserving interface fluxes are used at the element boundaries, together with the Strong Stability Preserving (SSP) time integrators, the cell-average solution remains positive under a positive time step size limit. Accordingly, with the Zhang-Shu limiter, or related approaches, the high-order solution inside each element can be corrected toward the cell-average solution to enforce pointwise positivity. However, these high-order positivity-preserving approaches are currently restricted in the literature to SSP-RK time integration schemes, thereby limiting the use of efficient non-SSP methods such as the classical fourth-order RK44 scheme. In this work, we analytically demonstrate that a certain class of non-SSP time integration schemes, including RK44, maintains positivity of cell-average solutions under positive time-step sizes when positivity-preserving interface fluxes are used. As a result, the Zhang-Shu limiting approach and related correction techniques can be applied to construct high-order positivity-preserving solutions with RK44 and a broad class of non-SSP time integration schemes. Numerical experiments will be explored to show that, as expected from theory, with the scale-resolving simulations cell-average solutions remain positive with RK44 and a series of other non-SSP schemes with nonzero time steps, and that the Zhang-Shu limiter allows constructing high-order positive solutions with these time integrators.