I make mine the way you do - but a friend of mine (it's her recipe) makes it the gunky way. I just left her recipe the way she wrote it.
Depends on how you like your rice - if you use leftover rice that has been refrigerated overnight, the steam from the heat of the pan and the liquid in the pan will re-steam and hydrate the rice making it very good; fresh cooked rice seems to be "gunkier" because of all the moisture in the pan.
i have tried to fry freshly cooked rice... the thing i do is open the lid of the rice cooker and placing it in front of a fan until it cools then mixing it so that the inside part would be cool also... maybe doing it for 30 minutes