The rationale behind "She is coming back next Wednesday" without "on"

It is a reasonable desire to find the rationale behind a grammatical rule.
It's what linguists want to do. It's what teachers and students would like to know.

Unfortunately, there are problems. One is that many rules don't have a rationale. They just are. Also, they are very rarely stated specifically. Even when they are, they're quite often wrong.

In this specific case, there can't be a hard rule, because dropping on with these phrases is an optional phenomenon. That is, the speaker may do it, or may not, at their sole discretion, depending on how they're feeling at the moment. Nobody can tell them whether to use it or not.

It is always allowed to use on with dates. As noted, determiners like next, last, this, that, some, every can precede the dates after on, and the on may be omitted. This is also true of any noun phrase that refers to a specific date:

• He returned home (on) the same day she left.
• I'm supposed to meet him (on) some day next month.

But it doesn't work with indefinite noun phrases denoting dates:

• I like to run a on day like this.
• *I like to run a day like this. (asterisk means ungrammatical)

As for the rationale, it's like a lot of deletions of small words in English.
Complementizer that, infinitive to, wh+be in Whiz-deletion, Conversational deletion, and so on. These are all meaningless words that guide the listener to the structures. As such, they are predictable.

When they're talking fast, with somebody they know well, people tend to leave things out. Every preposition in English is an unstressed syllable. English is a stress-timed language, in which unstressed syllables are reduced and jammed into the time between stressed syllables. So English speakers tend to delete predictable, unnecessary, unstressed syllables. A lot of these are prepositions. That's all, really.