The concept of irreversibility of a quantum measurement process is an important point. However even in classical mechanics if a body goes from some asymmetric state to a state of [unstable] equilibrium (e.g. a theoretically possible process of a ball having just enough velocity to roll uphill and stop at the very top, not even spontaneous symmetry breaking needed) - you can't always predict backward in time as good as forward (and sometimes the other way round).
I was going to say that this doesn't happen, something could not travel towards an unstable point of equilibrium and then stop there, because it is travelling
to that point and those travelling
at that point. But I now realise that you're counting the friction (or equivalent resistance) that takes the (possibly minute) final velocity and (eventually) squashes it entirely.
Except that the energy goes somewhere. Into the movement of the air, the nature of the ground it's rolling over, all kinds of ways. And if you had the power to temporally reflect all of these effects back again, they'd give the kick to the stationary object in the right direction.
Yes, it's rather contrived. Imagine choreographing a nearly concentric set of water-pressure waves to converge upon the location of a stone at the bottom of a pond, lift it up through the water column until it emerges at the surface, at which point further surface waves and a contracting hemisphere of air-pressure ridges fling the stone fully out of the water (shaking lose its hydrophilicly bonded damp surface contamination), up and over into a hand (drawing itself back as the stone arrives so that it can clasp and slow the trajectory back down to a stationary 'end-point'.
Not arguing with your point, which is about the ability to predict 'the wrong way'[1] in time's arrow, but I just like bringing such imaginings to life. Another example of which might regard fragments of porcelain which an inrush of air-compressions and other energies push together to meld into a cup and launch up into the air such that at the height of the parabola it barely slides onto the top of a table, to (after gaining a brief amount of vibrational energy that marginally increases it speed, until it nudges against, and is brought to a halt by a person's elbow.
[1] As you note, some things are easier to backtrack than to predict before they start, while other things are easier to predict than determine a starting situation that brought you to a resulting one. Either way, your main problem is not having infinitely precise positional information, nor the time and processing power needed to deal with such an infinite amount of precision. (And, of course, elements of understanding about the system being only understood to the current level of scientific knowledge.)