I think "organic" here means the weapons "belong" to the unit using them (rifles and 60mm mortars as opposed to F-18's and tanks for an infantry company).
I appreciate your explanation. But sometimes I feel that our language is being bogged down with euphemisms and terms-of-art that try much too hard to paint a happy face on the realities of life. I guess I prefer something more up-front and direct, like, "we use sniper fire to kill these son of bitches before they kill us", rather than, "we destroy targets with organic direct fire weapons".
It's just a matter of taste.
--Boot Hill