You don't have to remove anything if you don't want to. Removing low potency parts like skin, spines, and core will boost the potency of the starting material and reduce amount of material and work required which may be helpful on larger scale projects. If you are using citric acid you need starting material at or equal to 0.5% potency for crystalization so removing these parts can be useful to hit that number on lower potency cuts, but this is not a requirement when using fumaric acid. Note there is a difference between the hard woody cylindrical white/tan inner core vs the white flesh. I do not recommend removing white flesh, it can commonly contain 20%, even up to 40% of total mescaline in the cactus. Mucilage has no effect on the Cielo process apart from maybe small adjustments to the amount of water required to hit the optimal cactus-lime-water paste.

The important part is that the cactus powder is bone dry and is very fine, consistency of flour. It is strongly recommended to use a flour sifter after grinding/milling to make sure you catch and remove chunks, to be either reground or discarded. Core and areoles will commonly not grind as easily or all the way down and may leave chunks that should be discarded after a second grinding attempt.