Specualtion ofc, but i'll give it a shot:
Misato depends on Kaji's validation for motivation. When he dies she is unable to do anything and cnnsequently doesn't really do anything when she clearly needed to intervene with Asuka, who in turn Shinji relies on for validation rendering him insecure. When the fan stops spinning is when Misato tells him he doesn't have a reason to be insecure. Misato's telling Shinji how to break the cycle, so the fan stops spinning.
At the risk of being that guy... is it even necessary that fans have symbolism attached to them now?
The fan in Misato's room has a perfectly valid reason to be there: it's perpetual summer and really, really hot. That's why she's often in such light clothing outside of work. It gets tied in with the cross symbolism in End of Evangelion sure, but the original TV version doesn't do anything with the fan other than "it's there". Possibly just as an establishing detail of the room.
Sometimes a cigar is just a cigar.
Yeah see, this here is why I subscribe to death of the author. You don't need to prove or assume intention to analyze how a narrative element is developed over time. The fan is spinning and it spins in different scenes which use the devices of Misato and Kaji in different contexts. The fan itself changes over tie from being functional to not functioning as well as being seen with different angles.
In our own stories we can draw from this kind of development to give more for whoever, reds, watches, or listens to what we're making. What Anno intended doesn't really matter. The elements of the craft do. This is why we do analysis.