Check out this site for some kanji etymology. I've posted the results there.
産 : 生 (Type 2 Phonetic) birth; life + an abbreviated form of 彦 (handsome/well-formed) → (well-formed), locally grown, life-sustaining product → produce; production; local product/material (that sustains life) → one's birthplace → childbirth.
生 : SIS shows an abbreviated form (Type 8 Phonetic) of 艸 grass/plant + 土 earth → fresh, slender grass shoots → birth; life → new; fresh; sprout; raw; alive; by nature; youth; unripe; pure; bear fruit; spare a life; bring back to life; make use of; make a living; be/remain valid; arrange flowers/plants; grow; undiluted.
Funny enough I just saw these two kanji used in a short story by Kenzaburo Oe this evening while studying. In the story なぜ子供は学校に行かねばならないのか, the narrator uses the 産む when referring to his mother's speech, where 生む is used in the passive form (生まれる). Both are talking about giving birth, but its interesting to think about how the usage of 産 in this case does refer more explicitly to the production of a child.
Although that alone, even combined with the kanji etymology, isn't conclusive, but it seems to me like you're on the right path when viewing 産 as more related to production whereas 生 is inherently more related to life, as a general concept, e.g. 生物、[生]{い}きる、生活