I don't know that much about the insides of the injection pump, but I read that they use the fuel pressure inside the pump to work the timing advance inside the pump. If the fuel pressure is not consistent coming into the pump, then it affects the timing advance of the pump and it can cause the engine to surge and rpms to vary up and down.
You're right, but not for the reason you expect.
The stock lift pump only outputs a max of 7 PSI; I'm seeing 5 at idle. Even going down to 0(which the pump is rated for - gravity feed), that's only a 5 psi difference.
If the lift pump fails entirely, the engine will still run just fine, with perhaps a couple of degrees less advance - remember, the IP has a built in "transfer pump" that will pull fuel in and boost it up to a higher PSI for internal use.
However, if you have air coming in with the fuel... air is compressible. It does weird stuff to a hydraulic system like our pumps. Also, if the air gets in the injection lines and goes to the injectors, you get a second whammy, because the RPMs start to drop and surge due to, well, less fuel being injected.
The IP compensates by adding more fuel(more metering valve travel), and then it surges ahead because the air bubble was just a bubble, and now it's injecting more fuel.
The IP compensates, and the process repeats.
Also, I've noticed that IPs are more 'sensitive' governor wise when the cold advance is on and there is no housing pressure built up. Not entirely sure why this is, but I'm sure it has to do with pressure differentials somewhere.