Why does 他吃了饭 sound incomplete?
I can't for the life of me understand why switching an associative verb to the front forms a question in English. Am I not stupid? 

The fact is, in a lot of cases, languages don't follow any kind of logic or reasoning. 

If you say 他吃饭了, it's a complete sentence. If you say 他吃了饭, a listener would expect you to tell him what happened next, like 他吃了饭，拍拍肚子就走了。Now that it's complete, the listener can rest in peace.

So in this case, 他吃了饭 means "After he ate" -- Apparently an incomplete sentence. 

But why does switching the position of 了 change its meaning? No one has the slightest idea.

There are situations where "verb + 了 + object" makes a complete sentence. Like when answering "他吃了什么?" with "他吃了一只火鸡."