Disconnecting pull chain from ceiling fan to install Lutron 'Maestro' fan/light switch
Step 15 of these instructions suggest disconnecting (removing) the pull chain extentions, meaning the extra length of chain hanging down for convenience.  It doesn't suggest disconnecting the switch itself (i.e. no internal fan re-wiring is needed).

FWIW - Lutrons instructions are very easy to misinterpret in this step, I can see how you arrived at this point.