Please, why can’t I use "the below example"?

Adjectiving below is not always well received.

As I’ve written before, it’s because the below whatever is nearly always better written the following whatever or quite simply this whatever. It sounds foreign otherwise; it is rarely if ever used by native speakers, as native Englander Barrie England observes.

Courtesy of Darshan Chaudhary in a before comment on the original post, a nice alternative is to instead use in the example given below. Because that now uses below as a traditional adverb, it sounds just fine.

Using Syntactic Tests for Adjectives

You’ll find that below makes for a highly suspect adjective, because attempting to use it as one would use other adjectives readily produces sentences that are ungrammatical. We can use syntactic tests employing substitution of below for normal adjectives to see that it is not of their category.

Here are a couple of easy demonstrations:

1. How many illustrative examples were you considering?
   *How many below examples were you considering? [ungrammatical]

2. Please include a more illustrative example.
   *Please include a more below example. [ungrammatical]

3. This big red firetruck shows its age.
   *This big below firetruck shows its age. [ungrammatical]
   *This below red firetruck shows its age. [ungrammatical]
   

Given that we cannot swap below in for other adjectives in the sentences shown above, it doesn’t really feel like much of an adjective at all. That’s why it feels odd to attempt to use it in places where only adjectives go in a noun phrase such as in the examples you provided.

"The example below" is more common. If you replace "below" with "above" then both orders are very common. Still, I recommend "the example below," which sounds a bit more natural.
The primary reason someone would take 'offence' to 'the below example' is mainly because 'below' is a preposition or adverb, and using it in an instance like 'the below example' is setting it up to sound like an adjective, it isn't grammatically incorrect, but it is not considered fluent usage.

The only reason is because it sounds terrible to most native speakers of American English and British English. We don't use the word that way.

You can say it. Countless non-native speakers do. And they probably do so by analogy to the above, which represents a logical thinking process. However, as little as one generation ago the above was not acceptable to most usage panels of respected dictionaries. Today, however most usage panels accept, if not unanimously, the above. It's probably a matter of time, like if in one or two generations 2 billion native speakers in Asia still say the below. There are, after all, different Englishes. However, for now, just note that most native speakers of American English and British English do not use the below example and for many its usage is worse than hearing the screech of fingernails across a blackboard. This should teach you that what is grammatical is determined by speakers, and that that is often determined by what sounds okay, or not.