Finney was not a slinger.
Pretty much everything he ever said about slinging was completely wrong.
So, yeah, if that's your only reference, you will have problems
Good, hard and accurate underarm is done without using your shoulder in any significant way.
It's actually the best style to learn first, as it demonstrates what the sling can do with very little effort on the users part.
It's mainly wrist, with a little elbow, and like every other style, just another weapon in a slingers armoury.
Like the underhand in this, but done standing up
https://youtu.be/1xSdOtttLAoAlso the sidearm in that is pretty shoulder friendly.
And used with legs and waist, really good for power and accuracy.