Why is the satisfiability of ESO formulas not equal to the satisfiability of FO formulas?
You're right: an ESO sentence is satisfiable iff its first-order "matrix" - with relation/function variables replaced with corresponding relation/function symbols - is satisfiable. 

TI'm not an expert here, but I think what's going on is "linguistic convenience." There are other related questions where we don't get coincision. For example, checking whether a first-order sentence is true in a given finite structure is usually harder than checking whether one of its "second-orderizations" is true in that same structure${}$('s reduct). Similarly, as you say validity for ESO sentences is strictly more complicated than validity for FO sentences. In this way it may feel more consistent to talk about ESO satisfiability rather than FO satisfiability if we're also talking about ESO validity, ESO truth in a given structure, etc.