Gemara Shabbat Muktza

The Gemara on Shabbat 123 discusses the laws of Muktaza. It tells us that in the days of Nehemiah, they decreed a prohibition on shaking all vessels (additions exempt plates and cups), and over time they permitted it in several waves, until the laws of Muktaza of the Tannaim.
 
“Rabbi Elazar said: The kenin and the glostra and meducha sticks are all changed before the vessels are removed. Kenin – our law: Neither the arrangement of the kenin nor their removal [in the Temple, reeds were placed between the loaves of the Presence to prevent them from getting soggy] delays the Sabbath. Sticks – our law: Thin sticks, divided into pieces, were there and placed on his shoulder and on the shoulder of his companion, and he hung and untied them. (Said) Rabbi Elazar: Fourteen, who is sick on the Sabbath, he places his hand on the shoulder of his companion, and his companion’s hand on his shoulder, and he hung and untied them….Rabba said: From what? Is it not always a mother to you – after the vessels are removed, we are changed. Reeds are the reason for this – because of the fact that he is sick, because he does not get sick in the Purta. Sticks – it is possible, as Rabbi Elazar said.”
I don’t fully understand Rabba’s rejection. He is actually arguing that it is possible that the use of reeds/sticks was changed after the utensils were permitted, but that they did not specifically permit reeds/sticks, because one can do without them.
Does it follow from this that even with the tools that were permitted (and here we are talking about tools whose job is to permit) , the permit is only when there is a “need”? And what is the definition of such a need?
The Rashba, the Ritva, and the Toss of Rosh say that in the Kenim, since the search until the end of Shabbat is negligible, it is called shaking “from heat to shade.” But this excuse does not apply to the matter of the sticks in the Pesach context.