Entropy is the only thing that could explain time (as global change) having a direction.
So the problem for all of physics is that it is "mechanical" - an assumption of local time reversibility is built into the very form of its equations. Any equation is a statement of a symmetry which is then broken by you the observer plugging in some actual physical number of a variable.
To then talk about the whole of the Universe, including the emergence of time, you have to have equations that already encode some sort of temporal direction - a symmetry breaking.
Thermodynamics or statistical mechanics give you such a framework. It says if you let a system freely randomise, it will freely keep randomising until that randomness reaches its equilibrium maximum.
But a theory of the universe has to include not only the story of its initial conditions (the local material contents that get shifted around randomly) but also the reason for their being "the system", quantum or otherwise, that is the boundary conditions.
So ordinary thermodynamics - which likes to take a bounded spacetime for granted - is only half an answer, which is why ordinary entropy accounting doesn't get you too far here.
On the specific thing of quantum information loss (or decoherence, as I believe you likely mean) then yes, that is a useful model.
The general idea is that as things tend towards a cold residual fizzle of radiation, all particular local information has been erased and you only need two global measurements - the general temperature (or energy density) of the universe and its scale factor (how big it is in terms of the visible event horizon).
At the Heat Death, with both these at their maximum, time has pretty much halted in terms of meaningful change. Like gas particles in a flask, everything might still be in motion, but nothing new is happening in any physical sense.