I'd be shocked if it was the fluid. It's not necessary to use genuine Hyundai SP-3. Remember, Hyundai/Kia doesn't "make" hydraulic fluid, someone does a private label for them.
There are many suitable replacements for SP3 sold by Beck/Arnley, Idemitsu, Castrol, Valvoline, MAG1 to name a few. Triax Global mfgs a Full Synthetic ATF and is something the Kia dealership could use because it's compatible in all non CVT Kia/Hyundai transmissions.
And Fwiw, I've been using MaxLife for years in my Sedona without any issues whatsoever. The manual calls for sp3.
I wouldn't be surprised if the flush created a clog somewhere. (I personally wouldn't ever flush an older trans but that's another discussion) Take it back and have them diagnose it properly, since they were the last ones to touch it. But if it is indeed looking like a clog created by the flush, be prepared to have a fight on your hands with the dealership. They're not going to want to cover the cost of a tear down. And don't let them talk you into another flush, your problem could easily go from bad to worse.
(it's possible there's something going on in the torque converter or with the pump but it would seem just too coincidental, especially after/since you just did a flush)