While it may not feel correct to you, a half past does exist as a variant of half past. In my idiolect, it does not sound odd.
The NOW Corpus, which draws from "web-based newspapers from 2010 to the present time," has 494 occurrences of "half past" without the indefinite and only 5 occurrences of "a half past" with the indefinite.1
Here are some examples of the latter:
- Uncle Bill came home about a half past 10
- And it's a half past four and I'm shifting gear.
- Shortly after, around a half past ten, the ladies of Wild Flag hit the stage
- breathed his last a half past three o'clock
- every morning at around a half past seven
These may sound odd to you because your idiolect does not allow them, but they are certainly used and sound fine to some.
Assuming you're right that some speakers find the article "extremely odd" in these constructions, an interesting question is which English dialects allow the indefinite article and which ones don't. That, however, is a project for someone else.
1. The former number, 494, is slightly misleading since I was not able to purge non-temporal "half past" constructions. Regardless, the variant without the article is much more common than the one with it.