When you synch the carbs your are trying to develop a 'relationship' so that all four cylinders draw in the same amount of fuel/air mixture.
In a perfect world, all four intake valves would open the exact same amount to let the fuel/air mixture in, all four cylinders would have the exact same compression to draw the same amount of fuel/air mixture in, and all four exhaust valve would open the exact same amount to expel the burnt charge.
In reality, valve clearances change and piston rings wear - this all effects the ability of each cylinder to draw in the same amount of fuel/air as the other cylinders
Chance are - we don't want the butterflies to open the exact same amount. Each cylinder is different and that is the purpose of the synch to begin with - to try to get all four cylinders to draw in the same amount of fuel/air. (to compensate for these wear factors) The should result in a smoother running engine.
yes, I'm aware of all those things, so we're on the same page so far.
You keep mentioning "idle" - idle has nothing to do with it. The throttle cable connects directly to the the "master carb" (#4). You are adjusting the linkage of the other three carbs to try and match the master carb. It shouldn't matter if you set this relationship at idle, 2,000 rpm or even higher. Once you establish this relationship - it should not change, it is "linear". (Do your valves open more at 6,000 rpm compared to 1,000 rpm??? Does your compression change also at higher rpms ???) NO
But here's where we differ. Idle has everything to do with it. At idle is where the engine is most susceptible to minute adjustments of the butterfly valves. At idle the butterfly valves are almost completely closed, and due to the low rate of airflow the engine vacuum is relatively high. There's a small adjustment screw for each of the other 3 carbs, used to micro-adjust the openings of each butterfly so each cylinders vacuum is equal to the master carb cylinder.
Let's just say for the sake of argument that at idle the butterfly air flow opening is 0.010" or 0.020". If you adjust one butterfly by 0.001", you've increased the opening by 5-10%, that's a fairly significant adjustment, and you'll see a noticable vacuum change because of it. Now open the throttle, and say the butterflies are now open by 0.250", or even 0.500". That same 0.001" adjustment is now virtually nothing, since its 0.2-0.4%, and at running speed you probably won't even be able to measure it, much less notice it.
So when you say the throttle response is linear, I think you're saying the same thing as I am, a 0.001" difference at idle is a 0.001" difference at WOT, no argument there. But that 0.001" difference is practically meaningless to air flow at all other throttle settings other than idle, that's why I keep saying that idle is the point at which carb synch adjustments are typically performed, because that's where the adjustment is most meaningful and measurable. Like I said earlier, I've been doing these adjustments for 30 years, and I remember the instructions with the carb stix always stressed that the adjustments be performed at idle, which makes sense to me based on my comments above. If you wanted to, you could also slowly increase the throttle to check other RPMs, but that wasn't as important or sensitive to adjustments as idle is.
Finally, your suggestion that the engine behaves identically at idle, 1k, 2k, or 6k RPM is a bit oversimplified, and its not linear at all. The volumetric efficiency of any engine varies non-linearly with RPM, that's why torque curves are never flat, or even a straight line. The usable compression changes significantly over the RPM range, because the engine fills the cylinders with air differently at different RPMs (because the butterflies are restricting air flow to the desired amount based on the throttle setting). The mechanical compression ratio doesn't change with RPM, but at small throttle openings you may only be filling the cylinder with 10% of the air it could theoretically draw with no restriction. As you open the throttle, that number could increase to say, 50-60%. As you open the throttle further, and RPM increases, then the throttle restriction isn't what reduces the amount of air the cylinder can breathe in each cycle, its the speed of the intake cycle. As the pistons move faster and faster, the amount of time the valves are open decreases, so the amount of air that can be pulled in tends to diminish, and as a result the peak torque (peak volumetric efficiency) is always achieved at something lower than peak RPM.
Edit: I looked at the latest Carb Stick instructions to see if anything has changed over the years, and here is what they say about the procedure using the latest version of their tool:
10. Start the engine and bring it back up to its normal operating temperature. With the engine at the proper idle, adjust the carburetor adjustment screws so that the SyncPRO™ fluid columns are approximately the same height. Once this is accomplished you have completed your carburetor synchronization.