Why "exhume" but not "exter"?

As far as I can find, there's this set of words for burying things and digging them up:

inhume and inter, both meaning put into earth

disinter (and apparently disinhume) meaning unput into earth

exhume, meaning take out of earth

But exter is missing, which I find peculiar. After all, it's all just putting things in and taking things ex of terra or humus, so you would expect a full complement, but this does not happen. The same pattern may emerge in other words as well, though none come to mind at the moment. Is there a subtle linguistic reason for which constructions pass into usage, or is it just an accident of history?

It's just a pitfall of taking the English language from Latin. Many words with prefixes that have opposites (e.g., in ↔ ex, pro ↔ anti, pre ↔ post) did not get their complementary opposite word.

A good example is disgust. The prefix dis- has several opposites, but the word disgust (root gust, meaning taste or stomach) has no complementary opposite using the opposite prefix.