I agree with Sylvia, the stress is usually caused by dancers stuck in the rigid, untraditional, RSCDS mode of dancing.
These dancers have the strange idea that phrasing, covering, and style are important when this is not the tradition in social ceilidh dancing.
Incidently, while the RSCDS was correct in publishing the dance to a 9/8 slip jig, this is only one of many traditions since there is no official "Strip the Willow" tune. I have noticed that in some dances in the borders, the musicians may indicate the start of a new top couple by using a slow hornpipe (or fast schottische if you like). Some dance a two step, while others just step hop. A common popular tune for this is "Siemsa Bhierta" [pardon my spelling as I have never seen the words written but the sound is similar to |SIM-za VI-ER-ta|. Some claim that Jimmy Shand wrote it as he published it under the name "Blue Bell Polka."
I do not recommend the step hop style as it is very tyring, but a skip change [two step] can be a nice alternative to the running step.