Time reversal symmetry for Newton's second law with time-dependent potential

Let me try and write and answer here which I hope OP will find sufficiently satisfying. Though it is exceedingly common in physics to use arrows to specify transformations, this is exceedingly bad notation and can often lead to confusions about what it is we are really trying to do.

So to start with, we suppose that we have some function $x(t)$ which satisfies the differential equation $$ frac{d^2 x(t)}{dt^2}=-partial_x V(x(t),t) $$ with some initial condition and where $V(x,t)$ is some known function.

So, what is time reversal symmetry actually supposed to mean to us? Typically when we refer to symmetries, we mean them only in the sense of preserving the action of the system. This is the correct technical requirement for (continuous) symmetries to give us conserved Noether charges. Though discrete symmetries, of which time reversal is one, do not lead to conserved charges, the same language is usually applied. I'll also point out that this is not just being pedantic as while symmetries of the action are always symmetries of the associated Euler-Lagrange equations, the converse need not hold.

With all that out of the way, let's be specific here about what we mean by a symmetry of the equations of motion. A symmetry is some function of the solution $x(t)$ which returns to us some other function $x^prime(t)$ which also satisfies the same differential equation. That is, symmetries of the equations of motion map solutions to solutions. So let's write down the proposed map for time reversal symmetry. Given a solution $x(t)$, we want to check whether $x^prime(t)=x(-t)$ is also a solution.

So let's see what differential equation $x^prime(t)$ satisfies: $$ frac{d^2 x^prime(t)}{dt^2}=frac{d^2 x(-t)}{dt^2}=left(frac{d^2 x(t^prime)}{dt^{prime,2}}right)Bigg|_{t^prime=-t}=left(-partial_x V(x(t^prime),t^prime)right)Bigg|_{t^prime=-t}=-partial_x V(x^prime(t),-t) $$ where in the last step I have used $x(-t)=x^prime(t)$.

What we have found is that this $x^prime(t)$ does not satisfy the same differential equation as $x(t)$ unless $partial_x V(x,t)=partial_x V(x,-t)$. This is perhaps not unexpected: it's the statement that we have time reversal symmetry so long as our potential is time-reversal symmetric. We could have reached the same conclusion much faster and with less hassle by looking at the Lagrangian and noting that the kinetic term is time-reversal symmetric, so the only real way for the action to remain invariant is if the time dependence of $V$ is reversal symmetric (even function in $t$).