The 009/10/11 series saws deliver oil at 6-12 cm3/min (0.37-0.74 cu. in/min) at 7000 r.p.m.
It's output is controlled by the RPM but limited by the amount of flex in diaphragm so I doubt it could for example double it's output when connected to a bigger powerhead.
Lets assume it could reach 18 cm3/min, this would be enough for smaller logs but probably not enough for bigger logs.
I like my Aux oilers to at least be able to match the max oil delivery rate of the power head.
For a 120 cc saw like a 880 that is 38 cm3/min, while the 3120 is 54 cm3/min, much of this is wasted going around the nose which is why a 3120 still benefits from a Aux oiler.
What's wrong with a gravity feed?
On my bigger CSM I have the output valve controlled via a bicycle gear lever/cable. This enables me to switch it on/off and control the flow from nothing up to about 50 cm3/min from the operator position.