Why is "for" used in "for example/instance"?

I have been looking at the interpretation of the word "for" in the phrases "for example" and "for instance", but I cannot make sense of it. The Oxford Dictionary lists 12 uses of "for" as a preposition, but to me, the "for" in the aforementioned phrases do not seem to match any of them.

Are these phrases some type of exceptions, where we can only interpret them jointly? In other words, is the "for" in these phrases not literally acting as a "for" preposition? Is this perhaps a phrase which came up as a simplification of what could be a literal yet longer alternative, like in

I am a keen reader. To give you an example of this, last week alone I read fourteen books.

John Lawler:

There is no meaning of for in for example; it's a fixed phrase, descended from another fixed phrase (for the sake of example), which in turn is descended from a Latin fixed phrase (exemplī grātiā), which doesn't have a for. Instead grātiā, 'thanks, sake, grace' appears in the Ablative case (that's the long ā at the end of grātiā), and that has the same use as the for in English, just like the possessive case of exemplī has the same use as of in of example. So there's no point in looking up prepositions in the dictionary. Mostly they don't mean anything.

Let me add that the Oxford English Dictionary indicates that the phrase for example is used as a "sentence adverbial" equivalent to by way of illustration or as an example or instance, and notes to compare it with the similar for instance and to the similar Middle French pour exemple and par exemple.

Am early instance in English is

1584 King James VI & I Ess. Prentise Poesie sig. Liijv, As for exempill, ȝe man not say Then feir nocht Nor heir ocht.

In for example and for instance, the word for indicates purpose. Here are some comparable uses of for in this sense:

John Dillinger drove to the bank for the purpose of robbing it.

Don't eat those fancy potato chips. I bought them for dinner, not for snackin'.

The spoiler on sports cars is just for show. It doesn't serve any real aerodynamic purpose.

For a good time, call 555-5555. [To achieve the purpose of having a good time, call that phone number.]

I bought the ping-pong balls for the cat, not for you. [I bought the ping-pong balls to benefit the cat, for the cat's sake, for the purpose of entertaining the cat—not to benefit you, etc.]

Kenneth Leahy built this truck-tractor combo just for fun.

For example is a fixed phrase: it has a customary meaning as a whole phrase, namely, to introduce an example. But this should not suggest that the individual words don't figure into the meaning of the phrase. Notice that the phrase is easily varied to suit variations in meaning:

For more examples of money-laundering, see Adam Khan's Twitter feed.

It is idiomatic:

For example (idiom):

As an illustrative instance:

• Wear something simple; for example, a skirt and blouse.

(AHD)

For example:

You use for example to introduce and emphasize something which shows that something is true.

• ...'educational toys' that are designed to promote the development of, for example, children's spatial ability. Take, for example, the simple sentence: 'The man climbed up the hill'. A few simple precautions can be taken, for example ensuring that desks are the right height.

(Collins Dictionary)