A lot of things will temporarily soothe the throat. Generally, anything thick like slippery elm will coat your throat and relieve pain just because it is a soothing, cool touch, and not because of any medicinal properties.
Slippery elm is another so-called alternative medicine that has never been scientifically shown to have any useful medicinal properties. This doesn’t mean it doesn’t have any, but humans have been remarkably good at extracting real medicines from our own environment, and the lack of any real data on slippery elm is telling.
So, yes, the stuff will make you feel a little better for a while, but unless you are convinced enough of the elm’s efficacy to have a legitimate placebo effect, it really isn’t going to do anything for you. The fact that the FDA doesn’t regulate, approve or test supplements, combined with their typically high price, is enough to prevent me from experimenting with it on myself.