The basic premise is correct as aside from water cooling systems, all the heat loss from the PC is by transfer to the air inside the case. The greater the air volume, the greater the ability to absorb heat.
But then Yewen's point takes over - once the air has taken up all the heat it can, it needs to be replaced by new cool air.
In general, it seems to be better the have more fans pulling air into a case than pulling it out as the high pressure air leaks out of gaps in the case and 'helps' itself out through the outlet fans.
The classic big volume high airflow case is the Akasa Eclipse with Intel Side Panel. 2 x 120mm intake fans and 1 x 120mm exhaust fan in a huge air volume.
The Lian-Li V1000 takes a different, but effective tack - the single hottest component, the PSU - is in an air tunnel underneath the main case with a relatively conventional in-out air path for the main components.