Given an ode $x' = f(t)$. Then a basic Euler discretization scheme yields $$ x_{n+1} = x_n + h f(t_n).$$
Now suppose you have a delay differential equation, say $x' = f(t-\tau)$, does it make sense to discretize as follows: $$ x_{n+1} = x_n+ hf(t_n - \tau)?$$
We fix $\tau$, and we define the function g, such as $g(t)=f(t-\tau)$ we have to solve $x'(t)=g(t)$.
Discretize it with an Euler scheme ( with all the convergence issues that come with it), it is not different from your initial equation.