Why were the sticks in Pesachim 5:9 muktzah if they were designated to a use that was also needed on Shabbos?
Correct. The answer to your question is that the halachos of muktza went thru a metamorphosis, and there were different severities to the gezeira at different times. Originally since people weren’t careful about hilchos Shabbos, the halachos were very stringent, and almost all items were muktza. Later on when peopple starting being more careful about the halachos, the restrictions were eased until they took the form of the halachos that we have nowadays. The Gemora Shabbos 123b says that the reason why they could not use the sticks was because at that time, even such a stick was indeed muktza.