How the stabilizer cutout switch functionality has changed from Boeing 737 NG to 737 MAX?
What do you know, I found the answer to my exact question in the Seattle Times:
First of all, even after two fatal crashes,

Boeing declined to detail the specific functionality of the two switches.

Nevertheless,

The Seattle Times found that the left switch on the 737 NG model is capable of deactivating the buttons on the yoke that pilots regularly press with their thumb to control the horizontal stabilizer. The right switch on the 737 NG is capable of deactivating just the automated controls of the stabilizer.

So far this corresponds to the information I found in the 737 NG FCOM myself.

As Boeing was transitioning from its 737 NG model to the 737 MAX, the company altered the labeling and the purpose of those two switches.
On the newer 737 MAX, according to documents reviewed by The Times, those two switches were changed to perform the same function – flipping either one of them would turn off all electric controls of the stabilizer. That means there is no longer an option to turn off automated functions – such as MCAS – without also turning off the thumb buttons the pilots would normally use to control the stabilizer.

If the above is correct, then not only MCAS, but STS/autopilot as well cannot be turned off on the MAX while keeping electric trim on the control wheel functional.

Peter Lemme, a former Boeing flight-controls engineer, said if the company had maintained the switch design from the 737 NG, Boeing could have instructed pilots after the Lion Air crash last year to simply flip the “AUTO PILOT” switch to deactivate MCAS and continue flying with the normal trim buttons on the control wheel.

(^ bold is mine). This is exactly what I was alluding to in my question. I really want to hear the reasons for the change.

Lemme said he’s surprised that Boeing made the change to take away the functionality that could have allowed the pilots to shut off MCAS without shutting off the electric switches at their thumbs.

Oops.