If you want to get really technical, it's because the symmetries of the Minkowski metric are the Poincaré group. Which includes only rotations, translations and boosts, none of which correspond to acceleration. Meaning it's inherently impossible to make acceleration look like being stationary because of the geometry of spacetime.

If Alice flies by Bob at some relativistic speed, then there's a very simple coordinate transform (a Lorentz boost) that flips our perspective to Alice's pov; she's stationary and Bob is moving.

If Alice were to accelerate and we did the same thing, we'd end up with a "momentarily comoving reference frame," in which Alice is only "stationary" for an instant and Bob is moving at a constant speed as before. Or we could create a non-inertial reference frame which would look nothing like Bob's perspective, but Alice would be stationary.

Physics in non-inertial frames behaves differently, as a simple example: if stationary (or constant speed) Bob dropped an object while floating in space, it would remain there. If accelerating Alice tried the same thing, it would accelerate away from her. You can test this out in an accelerating car or train or whatever and see that it's fundamentally asymmetrical even before considering SR.

In terms of things like length contraction and time dilation, these are a little more complicated mathematically, but it's just an extension of the above asymmetry when spacetime is Minkowski rather than Euclidean. The difference in observed time is clear when looking at each person's worldline, Alice's isn't straight like Bob's and so she unambiguously experiences a different proper time and proper length.

Ultimately this means that even if Alice accelerates then passes Bob at a constantly speed, they'll both see one another's clocks running slow by the same amount, when Alice decelerates and returns to compare her stopwatch with Bob's they'll have very different totals which corresponds to how much time Alice lost during her acceleration.