There are quite a few community sites for FFXIV that have been running contests that aren't screenshot contests. Several sites just gave away a video card and signed PC CE, Gamer Escape just did a contest that let entrants choose their type of entry (screenshot, video, poem etc). So there are definitely non-screenshot contests out there, you just have to look for them

That having been said, (as also said above) screenshot contests are easy because anyone can join them. They aren't dependent on drawing ability or video editing skills etc. It's quick and easy.