With regard to the first question: "Should it not be at least the following? "

Boastful, promotional writing, e.g. ‘best ever’, ‘most popular’, etc., is off-putting.

The short answer is "yes." But a comma is also "usually used after such expressions as that is, namely, i.e., and e.g." (Chicago 14th Ed. 5.62). Also, since this is a guide for school in the U.S., I'd also say that the American standard for placing punctuation inside quotation marks (except in cases that may result in ambiguity) should be followed.

Thus, a minimalist correction would be:

Boastful, promotional writing, e.g., ‘best ever,’ ‘most popular,’ etc., is off-putting.

That is, indeed, a lot of punctuation, but it is now fully "correct." (This would be a good sentence to place on an editing test for employment or for school admissions for advanced degrees.)

With regard to the second question: "Could it be improved?"

Answer: Definitely. There are, however, a few problems that I notice with the suggested revision. 1) The explanatory phrase cannot 'stand alone' as it is, being a sentence fragment. 2) The classification for the largest problem with the sentence really has to do with something that many textbooks refer to as the "incomplete thought." We can fix the punctuation, and move the order of the phrases around to better or different effect, but to fix the underlying problem, we must be more intrusive and add the missing thoughts, words, connectors that would be required to "unpack" what is actually being said.

The first thing to notice is that the examples don't actually match the thing being talked about. "Promotional writing," as such, is not reducible to the mere use of a few typically indicative phrases such as "best ever" or "most popular." This idea that the use of such phrases is a common indicator of unwanted writing is itself the missing thought.

Streamlining the core message is, in my judgement, an excellent start.

Boastful, promotional writing is off-putting.

Now, how to add the clarification? In my mind, the complete thought is basically this:

Such writing is often characterized by the overuse of superlative forms of comparison, such as 'best ever' or 'most popular,' which should be avoided in all cases in which these terms are overstatements or exaggerations.

Can this clarification be shortened? Probably, but the overriding point is that the original problem sentence cannot be completely solved without an intrusive edit -- so intrusive as to require the addition of more words than probably the initial sentence itself contained.

For cases in which, only minimalist editorial correction is allowed or permissible, there are always going to be certain problems that cannot be fixed (completely).