I honestly don't know why infitesimals aren't widespread. It can basically have the same basis/justification can't it? But with the bonus of being more intuitive.

Indeed they are more intuitive, people like Newton and Leibniz invented/discovered calculus by thinking in terms of infinitesimals, but it took time to be made rigorous, in the XX century. By then network effects got we stuck with epsilons and deltas, given that was the approach made rigorous earlier, and broadly adopted, despite being more cumbersome.