Can insulation batts run on top of wall studs?
You're pretty much right. Having potential airflow between the insulation and the drywall does hamper the effectiveness of the insulation. Likewise, having the fiberglass exposed allows more airflow through it, which also impacts its effectiveness.

Now, the air in attics tends to be relatively still (except when that attic fan of yours is running, of course), so neither factor renders the insulation completely useless (after all, we regularly use loose fill insulation in attics), but it does mean the actual R value of the assembly is quite a bit lower than the rating on the fiberglass batts.