It is correct that present continuous is used somewhat more often in (Turkey) Turkish. Actually present continuous will be the first tense that this course teaches properly. Partly because, from a few aspects, it has the most straight forward construction for conjugations and partly because it is, as you say, used a lot. But for the first lessons we needed some verbs to be able to construct sentences. So we decided to introduce a limited number of verbs and teach their conjugations in present simple tense. However the present simple tense will be taught properly quite later on the tree.
You only need to label the accusative when it's a specific object, not a general action.
Çocuk et yer = the child eats meat (generally speaking, not this or that meat; (s)he does the action of meat eating; the kid is not a finicky eater who refuses to eat meat the meat served to her/him)
Çocuk eti yer = the child eats the meat (the specific meat, for example, the pile of döner on his/her plate.)