...concerns [that] any major conflict...
...[which is] an outgrowth...
over concerns = because of concerns
over introduces the basis for an emotion in this usage:
He is upset over the accident.
The phrase conflict-averse means "averse to conflict", i.e. wanting to avoid conflict.
"conflict-averse over concerns" is awkward.
over should be because or inasmuch as or any other phrase that indicates cause-and-effect.
... they are conflict-averse because they fear any major conflict
would cost the lives of their sole sons or daughters