There is nothing wrong with interpreting Einstein's relativity theory as saying that one moves with speed c through time, in one's own rest frame.
Alex you do agree that past-present-future is not universal or absolute, correct? Also, you do know that simultaneity is only "simultaneous" to exactly one state of motion relative to specified events; all other states of motion will not perceive simultaneity of events.
Since one person's past can be another's future, and since what appears to occur at the same instant for one observer can appear to be "spread out" over a series of moments for another observer, then time is not "absolute" in the Newtonian sense... It's all relative to the observer at hand. The rate of passage of time is unique to the individual, there is no universal "clock".
The rate of pass of events (through time) in your living room can be sped up or slowed down simply by increasing or decreasing your living room's rate of motion (through space) with respect to my living room (or somebody else's house, or another planet, or a distant galaxy; just pick a frame and adjust SR equation parameters accordingly). That's not to say that were you sitting on your sofa, and your living room suddenly shot out into space, that you'd notice any "funny business" with your grandfather clock. No noticeable change except for a few G-forces... unless of course you look at the clock on the wall in my living room; you'll notice a peculiarity... my clock ticks slower! Same goes for me however if I were to look at your clock...
So space and time are inevitably linked, whereby past, present, future, and simultaneity, are dependent on rate of displacement between spatial points and location (with respect to relative phenomenon). You've heard the expression, "the farther out in space you look, the further back in time you see." And you do know that time-dilation has been confirmed again and again by experiment after experiment and observation, correct? (Atomic clocks loaded on two airplanes, muon detection at sea level, high-precision clocks measuring time on two different floors of high-rise (GR prediction), etc...)
You must know these things if you are in any way familiar with the implications of SR/GR...(??) Perhaps the only thing we disagree on, is definition of the word "motion" when I say "motion through time"... which is a nit-pick-technicality based on misinterpretation of contextual meaning...(??)