What is the "fundamental" difference between ‘search’ and ‘seek’?

So why do human beings spend so much time playing? One reason is that we have time for leisure; animals have very little time to play as most of their life is spent sleeping and (2)________ food.

^{_{Source: Cambridge English, FIRST 1 (2014): Test 1, Reading and Use of English paper}}

The following options are given:

A. searching ; B. looking; C. seeking; D. gaining

The answer given in the book is C, which is fine by me, in fact in the EL&U archives I found this question: "Seek" vs."search"; the accepted answer with twelve upvotes says:

[…] However, they are really quite different. One fundamental difference is that the object of seek is the item you are trying to locate, whereas the object of search is the place you are looking in. […] Also, seek implies that the seeker knows that the item they are looking for does exist. Search has no such implication.

Which confirms the "correct" answer is seeking. The other reason for it being the only answer possible is that we normally use the preposition ‘for’ with the verb search.

1. ‘search food’ (NO)
2. ‘search for food’ (YES).

Which brings me back to the EL&U question. If ‘food’ is the object of seek, as stated in the accepted answer, and a fundamental difference between seek and search; why do native speakers tend to say search for food?

The above Google Ngram seems to strongly suggest that searching for food is perfectly grammatical.

• If native speakers say seek food, and search for food; what is the fundamental difference between the two?

Solution 1:

Since the object of search would be a location (not the thing searched for), "food" cannot be used on its own as the object, as you have noted.

Instead, when we say "search for food", the "for food" is a modifying clause to clarify what we are searching for. It needs "for" precisely because "the object of seek is the item you are trying to locate, whereas the object of search is the place you are looking in."

Notice that even with "for", the clarifying clause is not serving as the object of "search". This becomes more clear if we include an actual object. For example:

search the room for food

"the room" is a location and is the object of the search. "for food" is still only a modifying clause that clarifies what the search is for. It doesn't thereby become the object of "search". Therefore, "search" and "seek" are indeed still different in the way you originally noted.

Solution 2:

Fundamentally, it comes down to knowing what is out there.

You search for something that you know (or at least are highly confident) exists. You just don't know where it is.
You seek something that you hope exists.

This also explains the use of for with search. You are searching for something known - for a specific item, person or solution that you will recognise as soon as you see it. Thus, search for food, search for a pair of shoes, search for your car keys.

You seek an unknown. You don't know exactly what it is. Thus you seek a cure for cancer, seek world peace, seek happiness.

A couple of examples:
He searches for the truth.
He seeks the truth.

The former may be used when trying to differentiate the truth from falsehood. There is a true solution to some problem and it is being searched for.
The latter is more philosophical or metaphorical. The truth is out there and one day we may know what it is.

Similarly:
He searches for food.
He seeks food.

Here the former implies that a hungry person is hunting for something to eat. This could even be on the level of choosing a restaurant.

The latter implies a quest for sustenance, be it physical food or food for the mind.