Or does it? The piston speed increases when the engine speed increases, but does that mean it can suck in more air through the closed air valve?
I recently worked on a OM616 engine with a governor with no leaks at all. I did the vacuum test: removed air hose, pushed lever in stop position, connected vacuum gauge, released lever. The gauge showed slightly below 100 mbar of vacuum and that is what you should expect. The FSM says 70-90 mbar, see:
07.1 Diesel Injection System - OM616
and select 07-125 over there.
I can make pictures or a movie this saturday of that governor, for comparison.
If you connect a vacuum pump to the governor, and if there are no leaks, the vacuum can go as high as the pump can provide (usually around 800 mbar of vacuum). It will not damage the governor, you are doing the same thing as putting the engine into the stop position.
According to my information you cannot get the engine to run well with a leaky governor housing. A slow leak is acceptable, but we are talking half a minute/minute here.
You can replace the diaphragm yourself, it is very well described in the FSM. You do need a micrometer and you have to watch out for small parts.
(That is my hand)
Fixing the diaphragma to the control rod is the most difficult, but do-able.