How come ∆G ≤ 0 is used for spontaneity in electrochemistry, not ∆G ≤ W(other)?
Most likely the source is wrong, or it has purposfully omitted the correct equation to pseudosimplify a problem. It might be that an electrochemical reaction was under discussion, not a cell. A further possibility is pursued in  the appendix. I recommend adding a source where you came upon the quoted exchange to avoid a straw man argument; even though in this case  I have observed (and possessed) the misconception first hand.
One definition of spontaneity
I define the true spontaneity condition as such when there is a tendency toward net 'something different' (chemical reaction, expansion etc) to establish state 2 instead of some original state 1. An example could be a chemical reaction where there is a higher reaction extent than at equilibrium, so a net reverse reaction takes place.
You are correct that the true spontaneity equation (from Clausius's inequality) in the thermodynamic limit is, for net-constant pressure and temperature in the system,
$$\operatorname{d} G-\delta w_\pu{other}\leq 0.\tag1\label1$$
When electrochemical work is the only component besides expansion work, this implies (due to $|\operatorname{d} n_\pu{e}|=|\nu_\pu{e}|\operatorname{d}\xi$)
$$\Delta_\pu{r} G_{T,P} +|\nu_\pu{e}|FE\leq0\tag2\label2$$
where $\Delta_\pu{r} = \partial/\partial \xi$; here $\xi$ is extent of the reaction. The Greek $|\nu_\pu{e}|$ signifies an absolute value of the stoichiometric coefficient of an electron in some half-reaction. Equation $\eqref2$ assumes that only one net reaction occurs. The term $|\nu_\pu{e}|FE$
should be a good indicator of electrical work.
It might also be that they are discussing an electrochemical reaction, not the cell itself. We can have the process
$$\ce{Zn(sln) + Cu^2+(sln) -&gt; Cu(sln) + Zn^2+(sln)}\tag3\label3$$
without harnessing its electrical work via an external circuit. So $\operatorname{d} G\leq 0$ would hold for spontaneity.
Appendix: An alternative
A different definition of spontaneity might be in play. Namely, the term spontaneity is also used to mean a large enough standard equilibrium constant (especially in biochemistry).
$$RT\ln \frac1K = \Delta_\pu{r} G^\circ_{T,P} \tag4\label4$$
Equation $\eqref4$ is technically a definition of the standard equilibrium constant. When non-negative absolute temperatures are quaranteed, the LHS of equation $\eqref4$ will become non-positive for all $K$ big enough, i.e., $K\ge1$ . That implies
$$\Delta_\pu{r} G^\circ_{T,P} \leq 0. \tag5\label5$$
Note, however, that $K\ge1$ doesn't necessarily imply much about the reaction balance itself because the equilibrium constant comprises of activities (not concentrations), and because the stoichiometric coefficients in the denominator may drown out the coefficients in the numerator. (Equation $\eqref6$ assumes that surrounding fugacity is equal to an agreed fugacity in the standard state, denoted here and elsewhere by '$^\circ$'.)
$$K(\pu{in solution, solvent \ce{A}}) = \left[a(\ce{A})_\pu{eq}\right]^{\nu_\ce{A}}\prod_i \left[a(\ce{B_$i$})_\pu{eq}\right]^{\nu_i}. \tag6\label6$$
Reactants–products (other than solvent) are designated by $\ce{B_$i$}$. But still, for $K$ big enough (2nd definition of spontaneity), equation $\eqref5$ will hold by definition. Also keep in mind $\Delta_\pu{r} G^\circ_{T,P} \neq \Delta G$ (even their dimensions are different!). So the source is presumably still at fault for poor notation.