I read a nice bit of research I can't currently locate, about how split humans are about whether we move through time, or time moves over us. I think it tuned in on how people interpreted 'move this date forward', whether that meant speed up or delay. I would say this points to how moving in time or time moving, are fundamentally metaphors, & in physics we should seek more precise statements.
You are exactly wrong to say movement through time doesn't fit with Relativity. Minkowski's interpretation makes a direct analogy between the symmetry operations of moving in spatial dimensions, and time pictured as a dimension for symmetry operations. The spacetime manifold in GR can be pictured as already existing ahead of us like the past does behind, but our subjective experience is moving through spacetime with a substantial degree of continuity, which is to say symmetry between past & future in a specific sense. Or, as a presentist, you could say the algorithm linking past & future moves through the present.
What we have to reconcile with spacetime, is the thermodynamic arrow of time: the accrual of irreversible changes. GR is completely reversible, so you might expect no fundamental difference in character between past & future. QFT is very close to reversible, with the exception of deep inelastic scattering in Weak Field interactions, which fit with a higher CPT symmetry - it's widely thought the matter-antimatter assymetry derives from this. But, Weak interactions have very little impact on our experiences, mainly just beta decays. So that leaves a bigger question about time in our experiences. Which more deeply, requires a quantum-gravity theory, uniting GR's spacetime & the thermodynamic arrow of time which seems to emerge from quantum behaviour.
Rovelli's Loop Quantum Gravity is an example of attempting this, where a deeper reality of a quantum spin network selects between 'crowds' of possibilities while maintaining symmetries like The Principle Of Least Action over all. There's also Chirobelli's Purification Principle, picturing time as the spreading out of correlations, & attempting to find the simplest set of axioms GR & QFT can both be derived from. Deutsch & Marletto's Universal Constructor theory expands the classical logic of Turing Machines, to include objects as defined by sets of symmetry operations, & so constantly including counterfactuals about how they are arranged.
We have to account for what we see, which overall is a lot of persistence through time, with exceptions that we can understand through conservation laws. Those can be linked to continuous symmetries through Noether's theorem. QFT is our best theory, but it provides no account of where time comes from, it simply assumes it as background, & so far has only been able to integrate Special Relativity, ie flat spacetime. Relativity gives us a limit on the speed of propagation of information, and we can think of change as occuring in a 'channel' between isolated systems & speed of light, a ripple of information ground out by the laws of physics, in which sense there is no unified now, and there are no subjectivities to be moved past or who move through, there is only the ticking of light-clocks until only photons are left & time itself becomes meaningless (in Penrose's Conformal Cyclic Cosmology, which seems pretty compelling to me in accounting for the entropy assymetry at the Big Bang).