Have...going for one/oneself

My reaction upon reading every one of those counterexamples was, "That's obviously wrong!"

Have (something) going for you is and idiomatic expression from the early ‘60

If you have something going for you, or if something has something going for it, there are certain advantages that will make the end result successful:

• They've got a happy marriage, great careers, wonderful kids - in fact they've got everything going for them.

(Cambridge Dictionary)

Have (something) going for one:

to have a talent, skill, etc., that helps one

• She's not as young as some of the other athletes, but experience helps, and she has that going for her. You should be more confident in yourself. You have a lot going for you!

(M-W)

Usage as evidenced by Google Books is most often correct with only a few instances of the use of the reflexive pronoun and all main dictionaries don’t use definitions or usage examples with reflexive pronouns.

I’d avoid using the reflexive pronoun even though the meaning is clear, so we could say that its usage is non standard.

While I certainly concur with the other answers on this subject, and the OP’s own feelings, that the use seems to be non-standard, I think what possibly is happening in the usage "have something going for oneself" is the the emphasis on one, from some aspect, by being the only recipient of the advantage, etc.:

As in the definition of oneself in ODO:

2 [emphatic] Used to emphasize that one does something individually or unaided.

‘the idea of publishing a book oneself’

Or in Macmillan:

2 used for emphasizing that you and not anyone else does something

It’s important to complete the application forms oneself.

If we consider this emphasis to be part of the usage, then all examples listed in the OP, as well as any other usage makes sense and the non-standardness is somewhat resolved.

This emphasis point seems to be supported also by the fact that many of the Google search hits point to celebrities or people who are somehow are considered outstanding and unique, just one example of the several:

As a solo artist this Toronto Reggae Singer has a lot going for herself. She writes, records, produces and is the Musical Ambassador for Read Across Jamaica Foundation

With all that said, lacking this intentional emphasis, I am neither able to find any definition or usage note that would justify using oneself instead of one, so in such uses it would be unquestionably non-standard.

While common usage of the phrase probably suffers from inconsistency and sloppiness (I can see myself [me?] using either without thinking about it), I suppose there could be a distinction depending on what it was that was doing the "going for".

If the advantage was coming from within, then the reflexive form of the pronoun seems more apt. If the advantage is coming from outside, then the accusative seems right.

Ex:

1. The only thing she had going for herself was her determination.
2. The only thing she had going for her was her doting father.