Hmmm. Guess this old dog just learned a new trick. Though I've been shooting sticks for more than 50 years, I really haven't experimented too much with side plate materials; thickness, yes, materials, no. I've used mostly leather over the years, and switched to mostly velcro the past 10-15 years or so.
If your arrow is shooting stiff and you move the side plate out, it should shoot stiffer yet, but what you're saying is that a softer material will make it shoot weaker even if it's thicker.
I can envision the action that would have to occur to make that happen. The arrow tip/tip area, when it's released, does momentarily press into the bow side plate as it begins its first bend. Of course, the shaft quickly bends the other way as it moves forward such that the entire shaft clears the bow if it's spined correctly.
If the side plate material is so soft that it compresses more (than the compared material) at the initial moment of the release, the point of the arrow/arrow would be closer to the center of the bow at the side plate at release, which would require a bit more spine. Or, said another way, it would make the stiff spine shoot a little weaker, which is what happened in your case.
So, theoretically, it's possible. I find it hard to believe though that the difference in softness between velcro and a rug rest is great enough to cause that effect.
I think rifleman's explanation might be more plausible, or maybe it's the combination of the two. Regardless, glad it works for you.