Time reversal symmetry in the presence of friction

You have all the elements in your question, your difficulty is about what is meant by "time reversal symmetry". Time reversal symmetry holds if, when "playing backwards", the motion observed obeys the same law. With friction it is not the case : friction opposes movement, when playing backwards it (seemingly) promotes it.

You can also go to equations for this. Let's have a damped oscillator of mass $m$: $$ m \frac{\mathrm d^2x}{\mathrm dt^2} = - c \frac{\mathrm dx}{\mathrm dt} - k x $$ Now play backwards with $\tau=-t$, thus $\mathrm d\tau/\mathrm dt = -1$: $$ m \frac{\mathrm d^2x}{\mathrm d\tau^2} = c \frac{\mathrm dx}{\mathrm d\tau} - k x $$

So, the equation is not the same—unless $c= 0$ : the physics of the backward time phenomenon is not the same if there's friction—while it is the same when there is no friction.

The frictional force $always$ opposes the direction of motion. Try to pull a rock on a sandy beach, or swim in water. It does not matter what direction you move you experience resistance to the motion. So friction is $not$ reversible in the sense that you could 'reverse" it by playing it backwards in time, played forward or backward played, you have resistance.