Both the first linked video to the first version of the question(v1), and the second link video in later versions of the question, is about deriving Euler-Lagrange equations from the principle of stationary action.
1) Susskind does not mention time discretization in the first video. Time $t$ is there a continuous real parameter throughout the video, and the derivation is very similar to e.g. Chapter 2 in Herbert Goldstein, Classical Mechanics.
2) Susskind does indeed use time discretization in the new video. Many places in physics we can approximate a continuous model with a discrete lattice model, where derivatives get replaced with finite differences. In the end, we let the lattice spacing go to zero, and recover the continuous model. In the last chapter of his book, Goldstein speaks about transitions between discrete and continous models. This may help you to get the big picture. It is useful in physics to understand both the discrete and continuous approach. When done correctly, they should be equivalent, but for certain applications one approach is often more convenient than the other. I'll try to make a future update if I find an exact reference to Lenny's discretization argument.