Why is "president-elect" grammatically correct? Shouldn't it be "president-elected" or "elected president"?
President-elect is correct.
The word elect was appropriated to the English language in the 15th century from the Latin electus, the past participle of the verb eligere. In this form it acted as a noun or adjective designating the person chosen for an office, and that old use established the pattern reflected in President-elect.
Our verb elect arose about a hundred years later as a recategorization of the noun/adjective; if we had been concerned to return to the Latin root the verb would have been elige (which is reflected in the adjective eligible).
This, by the way, is very common in English etymologies. Many English verbs are formed on Latin past participles—almost all of our -ate words, for instance, such as demonstrate, aggregate, complicate, arbitrate.
Try to think of it as a stock phrase.
Some phrases of Latin or French origin 'require' the adjective to be placed after the noun.
I can give several additional examples here: secretary general, court martial, battle royal, heir apparent, devil incarnate, etc.
However, these uses are quite restricted. You'll have to learn them by rote.
Other phrases that may show the range of structure might include:
- District champion
- Preseason favorite [can be used later in the season, and we still use favorite instead of favorited/favored]
- Expected graduate
So the focus can be on what was done, the continuing current quality (as having completed the action), or what the future expectation/consequence will be. There are some better examples, I am sure, of each.
But I believe we tend towards using "president-elect" rather than "future president" or "elected president" because the attention is on the current. Future president would put the attention down the road more, and perhaps also fail to give a measure of the learning process needed in transitioning into being the president (and it also goes against a common superstition/respect of not claiming something until it actually is true, due to any unforeseen potholes and to allow the current office to complete their job). Elected beckons back more to the election than the person now or the challenge at hand. So it seems sharpest to say "president elect" rather than the just-as-legitimate "future president" or "elected president".