The Dirac delta function is a mathematical fiction. It doesn't look weird in your case because you didn't mix it with any other weird mathematical fictions. But

The Dirac delta function is a mathematical fiction. It doesn't look weird in your case because you didn't mix it with any other weird mathematical fictions. But maybe somebody could invent a system that violated conservation of energy by using Dirac deltas. That wouldn't mean the Dirac delta is a bad fiction, because if you actually tried to build it using your step function approximation it wouldn't work.