Is it always grammatical to omit "on" before days?

I've noticed in several online news articles lately that authors no longer use the preposition "on" for the specific day on which an event occurred. Take the following for example:

Despite impassioned pleas from several countries, the World Health Organization's annual assembly refused Monday to even discuss admitting Taiwan to the meeting, under pressure from China. Read the rest of the article here.

Is the omission of "on" in the given example grammatical?

It would be possible to construct a sentence where the word "on" is necessary to avoid awkwardness. It might be a bit contrived.

For example, if we are going to meet Monday, and I want to say that when we meet, we will discuss something that happened Wednesday, then it would be awkward to say, "We can discuss the events of Wednesday Monday," but "We can discuss the events of Wednesday on Monday," is all right.

If you were talking about a person whose name is "Friday", for example, this could also lead to cases. "We'll meet Friday" is unclear, but "We'll meet Friday on Tuesday" is clearer, but still odd. I would probably call the person "Mr. Friday" or "Ms. Friday", assuming I know their gender.

Aside from contrived examples such as these, I don't think there is a case where omitting the "on" is really problematic.