As the answer to the linked question states, original Turkish delight is made using only cornstarch.
I understand you're asking specifically about the recipe linked and how it differs from other recipes. I'll try to answer the following points:
Q: Why do some recipes call other gelling agents such as gelatin, next to cornstarch.
A: Like most thickeners, gelling agents, cornstarch is also susceptible to syneresis/weeping. Using other hydrocolloids helps preventing this. In Turkey, apart from traditional confectioneries, to cut down the syneresis, pectin is mainly used as the choice of additional gelling agent.
Q: Can gelatin alone be used to get the same texture, as the traditional Turkish Delight?
A: The answer is simply: "No". As gelatin will form a brittle gel, which is not really even close to texture of the gel that would be formed by gelatinized cornstarch.