Are sport drinks a good electrolyte supplementation for my child with gastroenteritis?

No. These generally have an excess amount of sugar that can actually worsen the process. I have occasionally used gateraid diluted half & half with water in older kids (>6m).I use pedialyte in most.
Not ideal. For small children, the electrolytes are too few and sugar too much in most sport drinks to us alone for gastro- compare with eg pedialyte, which can be used for a day or so. For older kids, most clear fluids are ok for a day or two, especially if eating a little. Infants who cannot take their usual milk or formula should see their doctor.