There are two types of multi-circuit T/S panels:
The first one has two conductors for each circuit, one supply and one 'return', each circuit's switch is an SPDT type, and the circuit uses the original breakers in normal mode, and the T/S's breaker in generator mode.
The other has a single pair of conductors for the normal power supply, like a sub-panel, and the T/S's breakers are used in both normal and generator modes. Sometimes, the home runs are rerouted to the T/S.
Both types have a single neutral conductor and an EGC conductor, and you must maintain opposing phases when supplying MWBC's, but supplying only one conductor is no different than turning off only one breaker of a MWBC.
Either way, as long as everything is wired properly, there is no electrical reason to bring non-necessary conductors in or through the T/S's enclosure or raceway. BTW, I have never seen any instructions stating to do so.
"We've always done it that way!"