It is only great airflow when you run caseless if you have some sort of fan pulling air across the motherboard. If you run a motherboard with a hot chipset and you have your PC caseless, you still need some sort of air movement to keep the chipset cool. Part of the benfit of a case is that the fans should be setup to provide some airflow across the chipset to remove the heat. Yes, you might be exhausting that air back into your arcade cabinet but you are still moving heat away from the major heat producing devices.
In the end, if you aren't getting frequent computer lockups, then you have sufficient cooling. One other consideration. If you have small children (or young siblings, etc.) and your cabinet door doesn't have a lock on it, running a PC caseless is about as smart as licking a public toilet seat.
Dave